Computer system and method for pricing financial and insurance risks with historically-known or computer-generated probability distributions

ABSTRACT

The invention is a computer-implemented system and method, and a computer-readable medium for use with computer means, that enables portfolio managers to price, on a risk-adjusted basis, any traded or under written risk vehicle in finance and insurance that has a historically-known or computer-generated probability distribution. More importantly, the invention provides a universal approach to pricing assets and liabilities traded on an exchange or over-the-counter market, or underwritten for direct risk-transfer, even if those assets and liabilities are grouped or segregated, or whose prospective outcomes may alternate between positive or negative values.

FIELD OF THE INVENTION

The invention provides a universal approach to pricing assets andliabilities, whether traded on an exchange or over-the-counter market,or underwritten for direct risk-transfer, whether grouped or segregated,or even positive or negative in value. More particularly, it relates toa computer-implemented system and method, and a computer-readable mediumfor use with a computer means, for pricing financial and insurance riskswith historically-known or computer-generated probability distributions.The method can also be used to quantify the uncertainty of a variablewithin any historically-known or computer-generated distribution ofoutcomes. The invention can also be used to objectively assess therelative value of traded portfolios, and to assist the underwritingselection of managed accounts.

BACKGROUND OF THE INVENTION

An economic enterprise, particularly a financial firm, insurancecompany, or government agency, often faces uncertainty in the futurefinancial value of its assets and liabilities. These assets andliabilities can be brought to the enterprise via financial trading, orvia financial or insurance underwriting, and then managed within aportfolio. In the prior art, the uncertainty of these assets andliabilities were evaluated differently, based on whether they weretraded financial instruments, in a trading risk management environment,or, based on whether they were underwritten financial obligations, in anunderwriting risk environment.

Pricing of Assets and Liabilities within a Trading Environment

When assets and liabilities are managed within a trading environment,such as where stocks, bonds, currencies, commodities, or other financialinstruments are exchanged, uncertainty can be expressed as a probabilitydistribution of potential future market prices. For each instrument, aprobability distribution assigns a probability to each potential futureprice, as a potential outcome. Some outcomes increase, and othersdecrease, the future value of a portfolio that holds, and trades, thesefinancial instruments.

For example, a mutual fund trader faces uncertainties in a stockportfolio because of volatilities in the underlying stock price. Amunicipal bond issuer from city government faces uncertainties inrisk-free interest rates. A corporate treasurer faces uncertainties instrike prices for options issued to employees. A farmer facesuncertainties in soybean commodity futures prices before harvest time.

For each trading portfolio, a probability distribution of future pricescan be assigned to each individual instrument, or, to any grouping ofinstruments, or, to the entire portfolio of instruments. The shape,skew, and other aspects of this probability distribution are fitted tohistorical records of past price movements for those instruments. This“fitted distribution” can then be used in quantitative models toanticipate future price movements.

Historical data for the price changes of stocks, bonds, currencies,commodities, and other financial instruments can be fitted to differentkinds of probability distributions. These include parametricdistributions generated by a simple mathematical function, such asnormal, lognormal, gamma, Weibull, and Pareto distributions, andnon-parametric distributions generated from a set of mathematicalvalues, like those known from historical tables, or those generated fromcomputer simulations.

In the prior art, computer-implemented systems and methods, andcomputer-readable media for use with computer means, for pricingfinancial instruments, were deficient because they could not accuratelyprice, in a risk-neutral way, the vast majority of financial instrumentswhose price changes did not fit normal or lognormal probabilitydistributions.

Pricing of Assets and Liabilities within an Underwriting Environment

Assets and liabilities can be managed within an underwritingenvironment, where, for example, credit, health care, pension,insurance, and other risks are assumed. Uncertainty can be expressed asa probability distribution of anticipated contract obligations. Aprobability distribution assigns a probability to each contractobligation, as an outcome. Some outcomes increase, and others decrease,the future value of a portfolio holding these obligations.

For example, a credit card issuer faces uncertainties because ofcustomer delinquencies, defaults, renewals, prepayments, andfluctuations in outstanding balances. A utility company facesuncertainties in energy demand during extreme weather conditions. Ahospital faces uncertainties in patient receivables. An insurancecompany faces uncertainties in premium receptions and claim payments. Areinsurer faces uncertainties of paying for hurricane and earthquakedamages. A pension plan faces uncertainties of prolonged lifeexpectancy.

For each underwritten portfolio, a probability distribution ofanticipated obligations can be assigned to each individual contract, orto any collection of contracts, or to the entire portfolio of contracts.A parametric probability distribution can be fitted to historicalrecords of past experience for those contract obligations. This “fitteddistribution” can then be used in quantitative models to anticipatefuture contract obligations.

Historical data for contract obligations in credit, health care,pension, insurance, and other underwritten risks have been fitted todifferent kinds of probability distributions. These include parametricdistributions generated by a simple mathematical function, such asnormal, lognormal, gamma, Weibull, and Pareto distributions, andnon-parametric distributions generated from a set of mathematicalvalues, like those known from historical tables, or those generated fromcomputer simulations.

In the prior art, computer-implemented systems and methods, andcomputer-readable media for use with computer means, were deficientbecause they could not accurately price, in a risk-neutral way, the vastmajority of underwritten contract obligations whose cashflow outcomesdid not fit normal or lognormal probability distributions. This was trueeven when portfolios were expressly underwritten for immediate transferto another counterparty by true sale, trade, or even reinsurance.

Pricing of Risk Vehicles, Regardless of Whether they are Assets orLiabilities, Traded or Underwritten

In the prior art, computer-implemented systems and methods, andcomputer-readable media for use with computer means, were deficientbecause they could not accurately price, in a risk-neutral way, the vastmajority of financial instruments whose price changes did not fit normalor lognormal probability distributions.

Before this invention, computer-implemented systems and methods, andcomputer-readable media for use with computer means, were deficientbecause they could not accurately price, in a risk-neutral way, anymanaged portfolio of assets and liabilities whose entirety or parts weredrifting from positive to negative value, or, from negative to positivevalue, over time.

With the explosion in computer applications, reams of historical datafor financial instruments and contract obligations can now be gatheredand processed instantly. Many computer simulation models can generate asample distribution of possible outcomes for these traded andunderwritten portfolios. For example, a derivative modeling firm cananticipate a price distribution for an underlying stock for a financialoption. A catastrophe-modeling firm can anticipate a loss distributionfor a geographic area after a simulated hurricane or earthquake.

Yet, with the increased availability of historically-known orcomputer-generated data, there is no accurate method for pricing theunderlying risk, except in two special cases, applicable only to rareinstances of probability distributions.

The first special case is the well-known Capital Asset Pricing Model, orCAPM, which relates the expected rate of return to the standarddeviation of the rate of return. A standard assumption underlying theCAPM is that asset price movements have lognormal distributions, or,equivalently, that the rates of return for those asset price movementshave normal distributions. The CAPM approach is deficient, however, whenthe historical asset returns do not have normal distributions.

The second special case is the Nobel Prize winning Black-Scholes formulafor pricing options. Financial trading and insurance underwritingresearchers have noted the similarity in the payoff function between afinancial option and a stop-loss insurance cover. Again, a standardassumption underlying the Black-Scholes formula is that asset pricemovements have normal or lognormal distributions. Again, theBlack-Scholes approach is deficient since the historical price movementsof most capital assets do not have lognormal distributions.

To summarize, the historical data for traded and underwritten outcomesfor assets and liabilities rarely resembles a normal or lognormaldistribution. Most of the historical data fits other types ofprobability distributions. Most of the real-world traded andunderwritten outcomes therefore cannot be effectively priced in arisk-neutral way by current valuation models, including those based onCAPM, Black-Scholes, or other implementations of modern options pricingtheory.

There is a demand for a computer-implemented system and method, and acomputer-readable medium for use with computer means, to effectivelyprice all kinds of assets and liabilities, whether traded orunderwritten, grouped or segregated, mixed or homogenized, in variousand sundry ways, and whose probability distributions of uncertainoutcomes, for any positive or negative values, at any level of detail,may be fitted, to any parametric type, including, but not limited to,normal, lognormal, gamma, Weibull, and Pareto distributions, as well asany nonparametric type, generated from any set of mathematical values,like from a computer.

In the United States in particular, the deregulation of banking,securities, and insurance, will encourage the integration of differentportfolios of assets and liabilities, requiring such a unified approach.

GLOSSARY

Adjustment for Risk

Risk refers to potential deviations of cashflow outcome from an expectedmean value. An asset is defined as having a positive expected meanvalue. When evaluating a fair value for an asset, a prudent individualadjusts an asset for risk by inflating the probability for the worstoutcomes and deflating the probability for the best outcomes. Withadjusted new probabilities, the probability-weighted positive value forthe asset is reduced.

A liability is defined as having a negative expected mean value. Whenevaluating a fair value for a liability, a prudent individual adjusts aliability for risk by inflating the probability for the largest lossesand deflating the probability for the lowest losses. With adjusted newprobabilities, the probability-weighted negative value for the liabilityis increased.

Catastrophe Bond

A bond whose scheduled coupon or principal payments may be reduced inthe event of a catastrophe. If the yield is high enough, an investor maybe attracted to the bond, despite its default risk.

Contingent Payoff

A financial payoff whose value at least partly depends on the futurevalue for an underlying risk vehicle. For example, a call option is afinancial payoff whose value depends on the resulting price of anunderlying stock at the end of the life of the option. One purpose ofthe present invention is to find a price for a contingent payoffsuperposed upon the underlying variable X, as shown in FIG. 3, step 301,for a traded risk vehicle, and as shown in FIG. 5, step 501, for anunderwritten risk vehicle.

Cumulative Probabilities

A more technical name for the individual probability weights that, aftera lowest-to-highest sort of cashflow outcomes, are cumulated, one byone, in ascending order, so that the first cumulation is the individualprobability weight for the lowest cashflow outcome, and the secondcumulation is the combination of the individual probability weights forthe two lowest cashflow outcomes, and so on, until the last cumulationis the combination of the individual probability weights for all of thecashflow outcomes.

Concerning the uses, applications, and properties of the presentinvention, consider that, for any specific data value y of a variable X,the cumulative distribution function F(y) gives the probability that theoutcome of variable X will be less than or equal to y. Consider anascending sequence of all possible outcomes {x₁, x₂, . . . , x_(N)} withindividual probability weights {f(x₁), f(x₂), . . . , f(x_(N))},respectively. Now compute cumulative probabilities as follows:F(x_(n))=f(x₁)+f(x₂)+ . . . +f(x_(n)), for n=1, 2, . . . , N. Cumulativeprobabilities are produced in FIG. 2, step 205; FIG. 3, step 305; FIG.4, step 405; FIG. 5, step 505.

Decumulation of Probability Weights

In the computer-implemented system and method, and computer-readablemedium for use with computer means, of the invention, the new,transformed probability weights emerging from the Wang Transform kernelall need to be decumulated back into individual probability weights, asshown in FIG. 2, step 208; FIG. 3, step 308; FIG. 4, step 408; and FIG.5, step 508.

Derivative

A financial instrument whose value is partly or wholly derived from thebehavior, or value of, a referenced underlying financial instrument.

Discounting

In the computer-implemented system and method, and computer-readablemedium for use with computer means, of the invention, the undiscountedWang Price needs to be discounted by the risk-free interest rate to thefinal Wang Price, as shown in FIG. 2, step 211; FIG. 3, steps 311 or318, FIG. 4, step 412; and FIG. 5, steps 511 and 518.

Distribution

A general term for a probability distribution of future potentialcashflow outcomes for an underlying risk vehicle.

Distorted Probability Weights

A general term for probability weights that have been decumulated afterundergoing the core process, or kernel, of the Wang Transform. Theprobability weights have been transformed, or distorted, by the WangTransform. When the prospective future cashflow values of a risk vehicleare multiplied to these distorted weights, and their products summed,the resulting Wang Price reflects the adjustment for risk.

Empirical Distribution

A probability distribution that has been shaped by the data of pasthistorical experience.

Exceedence Probability

The overall probability that the future value of an underlying riskvehicle will exceed a certain amount of money, in terms of either gainsas an asset, or losses as a liability. Exceedence probabilities are usedfrequently in underwriting financial obligations in insurance, credit,health care, and pensions, and especially in catastrophe insurancepricing.

In more technical discussions of the uses, applications, and propertiesof the present invention, consider that, for any data value y of avariable X, the exceedence probability G(y) gives the probability thatthe outcome of variable X will exceed y. Note that G(y)=1−F(y).

Fair Value

A price for an asset or liability that has been adjusted for risk, anddiscounted by the risk-free interest rate.

Historical Distribution

A probability distribution that has been shaped by the data of pasthistorical experience.

Implied Lambda

The inferred “market price of risk” for a known probability distributionand current market price for an underlying risk vehicle, as calculatedby the computer-implemented method, and computer-readable medium for usewith a computer means, of the invention. For the very limited case wherean underlying risk vehicle has historically-known traded market pricesthat are lognormally distributed, the market price of risk over aspecific period of time equals the difference between the expected rateof return for the underlying risk vehicle, otherwise called mu, and therisk-free interest rate, otherwise called r, whose remainder is thendivided by the volatility of the return, where the volatility iscalculated as the standard deviation of the return, otherwise calledsigma.

Lambda can be implied from a known probability distribution and knowncurrent market price for an underlying risk vehicle by following all ofthe steps of FIG. 4.

Implied Volatility

Under the Nobel Prize winning Black-Scholes formula, there is aone-to-one correspondence between the price for a specified option andthe volatility of the underlying asset price. When prices for aspecified option and current underlying asset are both known, thecurrent volatility for that underlying can be inferred. This inferredvolatility is called implied volatility. Calculating an accurate impliedvolatility, however, relies heavily on the Black-Scholes assumption thatprice changes in the underlying asset are strictly lognormal indistribution.

Individual Probability Weights

A more technical name for the initial probabilities that are attached tocash value outcomes. For example, the price for PQR stock may have a 5%chance of gaining 20 dollars during the next three months, a 80% chanceof neither gaining or losing in value, and a 15% chance of losing 20dollars in value. These probabilities are considered to be weights,because, after the application of the Wang Transform kernel, their newlytransformed probabilities are multiplied to their respective cashflowvalues.

In more technical discussions of the uses, applications, and propertiesof the present invention, consider that, the probability f(x) isassigned to a specific cash value outcome x. These individualprobability weights can be found in FIGS. 2, 3, 4, and 5, in steps 203,303, 403, 503.

Iteration

Testing different lambda values so that the resulting Wang Price afterdiscounting by the risk free interest rate will match the observedmarket price for the underlying asset or liability in question.

Kernel

A core process in a computer-implemented method, or computer-readablemedium for use with computer means, of an invention. For this invention,the kernel, or core process, is the data-processing steps of the WangTransform, as shown in FIG. 1. For the two-factor model, the kernel, orcore process, is the data processing steps of the Wang Transform, asshown in FIG. 6.

Lambda

The market price of risk for an underlying risk vehicle, or for itscontingent payoff, which can be either selected for use in thecomputer-implemented system and method, and computer-readable medium foruse with computer means, of the invention, or implied from thecomputer-implemented method and computer-readable medium of theinvention.

When selected for use in the computer-implemented system and method, andcomputer-readable medium for use with computer means, of the invention,lambda is combined with the cumulative probability distribution, tocreate a “shift,” that is, a marked increase or decrease, in theirvalues. The “lambda shift” is shown in FIG. 1, step 104.

Market Price of Risk

A specific parameter value used in the computer-implemented system andmethod, and computer-readable medium for use with computer means, of theinvention, represented by “lambda,” a Greek letter.

For the very limited case of underlying risk vehicles whosehistorically-known traded market prices are lognormally distributed, themarket price of risk over a specific period of time equals thedifference between the expected rate of return for the underlying riskvehicle, and the risk-free interest rate, whose remainder is thendivided by the volatility of the return, where the volatility iscalculated as the standard deviation of the return.

In the real-world cases where an underlying risk vehicle exhibits anyprobability distribution, and any current market price,computer-implemented system and method, and computer-readable medium foruse with computer means, of the invention, can be used to infer themarket price of risk as a specific parameter value, in a way thatreplicates the result, but not the approach, of implementations ofmodern options theory for lognormal distributions.

Thus the computer-implemented system and method, and computer-readablemedium for use with computer means, of the invention, extends theapplicability of the market price of risk to any underwritten or tradedunderlying risk vehicle with any kind of historically-known orcomputer-generated probability distributions.

In the computer-implemented system and method, and computer-readablemedium for use with computer means, of the invention, an initial lambdavalue can be selected to test whether a known probability distributionproduces the known current underlying price. If not, this initial lambdavalue can be tweaked, by adjustment higher or lower, as shown in FIG. 3,step 314, and in FIG. 5, step 514.

Net Present Value

The discounted value of an underlying risk vehicle, from its futurevalue to the present, using the risk-free interest rate.

New Probability Weights

Old probability weights, in cumulative distributed form, are given tothe computer-implemented system and method, and computer-readable mediumfor use with computer means, of the invention, to produce newprobability weights, in cumulative distributed form, as shown in FIG. 2,step 207, FIG. 3, step 307; FIG. 4, step 407, and FIG. 5, step 507.

Normal Inversion

The inverse mapping from each cumulative probability to a correspondingoutcome of a normally distributed variable, with a mean equal to zero,and a variance equal to one.

In the computer-implemented system and method, and computer-readablemedium for use with computer means, of the invention, the standardnormal inversion is employed to inversely map a set of cumulatedprobabilities, before a selected lambda is added, and a standard normalcumulative distribution is performed, as shown on FIG. 1, step 103.

Parameter Uncertainty

At the tails of a projected probability distribution, a lambda shift maynot sufficiently incorporate the full adjustment needed to compensatefor increased uncertainty. Outlying and extreme events, by their nature,are not easily comparable to or hedgeable against mainstream projectedoutcomes.

For example, for way-out-of-the-money contingent claims, catastrophicinsurance losses, or way-beyond-a-horizon-date reinsurance claimsettlements, markets may be illiquid, benchmark data sparse,negotiations difficult, and the cost of keeping capital reserves high.These factors contribute to a special level of uncertainty associatedwith these extreme projected outcomes and their respectiveprobabilities.

Parameter uncertainty can be accounted for by using a two-factor model,as shown in FIG. 6.

Payoff Function

In the computer-implemented system and method, and computer-readablemedium for use with computer means, of the invention, a payoff functionis any customized table of outcomes that has been arranged between twocounterparties, consisting of one or more potential future events, andtheir respective cashflow value outcomes. To price the risk of thispayoff function, the counterparties will typically assess theprobabilities that each potential future event, and thus theirrespective cashflow value outcomes, will occur. A payoff function can bebased on an option, a stop-loss provision, or any other type ofcontingent event arrangement for transferring cashflows.

A payoff function is handled the same way as an underlying risk vehicle,in the computer-implemented system and method, and computer-readablemedium for use with computer means, of the invention, that is, a seriesof cashflow values are multiplied to their respective probabilityweights after they have been transformed, except that these cashflowvalues have been generated by applying the payoff function to eachvariable outcome of the underlying risk vehicle, as shown in FIG. 3,step 315, FIG. 4, step 409, and FIG. 5, step 515.

P-Measure

The “objective” probability weights and attached cash values for thefuture price of an underlying risk vehicle, as provided by ahistorically-known or computer-generated distribution. For example, thehistorical volatility of an underlying stock is traditionally defined asa P-measure. Underwriters typically use P-measure to describe risk. Inthe prior art, P-measures and Q-measures were hard to compare, likeapples and oranges, and almost impossible to translate. Thecomputer-implemented system and method, and computer-readable medium foruse with computer means, of the invention, transforms any type ofP-measure for an underlying risk vehicle into an equivalent Q-measure,with some adjustments to the market price of risk, allowing for sometranslation between the two measures.

Probability Distribution

In the computer-implemented system and method, and computer-readablemedium for use with computer means, of the invention, any table ofpotential future outcomes, comprised of two columns of numericinformation, the first column holding probabilities, or chances, of acash value occurring, and the second column holding the cash valuesthemselves. This “discretized” probability distribution, that is, a“row-by-row” set of outcome probabilities and cash values, can beselected from historically-known, or computer-generated prices. Such aprobability distribution is selected in FIG. 2, step 203; FIG. 3, step303; FIG. 4, step 403; and FIG. 5, step 503.

Q-Measure

The “subjective” probability weights and attached cash values for thefuture price of an underlying risk vehicle, as inferred or implied bymeans of isolating variables, decomposed from equations, that analyzethe current market price. For example, the implied volatility of anunderlying stock, as inferred from modern options theory, istraditionally defined as a Q-measure. Traders typically use Q-measure todescribe risk. In the prior art, P-measures and Q-measures were hard tocompare, like apples and oranges, and almost impossible to translate.The computer-implemented system and method, and computer-readable mediumfor use with computer means, of the invention, transforms any type ofP-measure for an underlying risk vehicle into an equivalent Q-measure,with some adjustments to the market price of risk, allowing for sometranslation.

Risk

Any exposure to uncertainty of cashflow outcome. Uncertain cashflowoutcomes for actual and potential commitments, as related to events,rights and obligations, each uniquely projecting an uncertain rate ofreturn. The collection of all potential deviations of value from thestatistical mean of an expectation.

Risk-Free Interest Rate

Yield in the U.S. Treasury bill or the U.S. Government bond for the sametime horizon in question.

Risk Management Environment

A collection of people, processes, and tools that identify, monitor,acquire, and dispose of risks, by means of underwriting, capitalizing,reserving, and transferring those risks.

Risk-Neutral

After adjusting an asset or liability for risk by inflating theprobabilities for the worst cashflow outcomes and deflating theprobabilities for the best outcomes, an individual evaluates an asset orliability using the expected mean value under the new probabilityweights. The new probability weights are called risk-neutralprobabilities.

Sort

In a sort, each pairing of outcome probability and cash value isarranged, when all cash values having the same sign, inlowest-to-highest absolute value order, that is, in ascending order.When the cash values have different signs, that is, prospectivelynegative and positive values, the cashflows are sorted fromworst-to-best, from the perspective of a holder of the risk vehicle.Each outcome probability “tags along” with its paired cash value duringthe sort. The sort is found in FIG. 2, step 204; FIG. 3, step 304; FIG.4, step 404; and FIG. 5, step 504.

Standard Normal Distribution

A welcome mathematical probability distribution with mean 0 and variance1.

Student-T Distribution

A mathematical probability distribution with a parameter k being thedegrees of freedom, where k can be any positive integer such as 3, 4, 5,etc. The degrees of freedom k can also be generalized to positivenon-integer numbers. The Student-t distribution can also be re-scaled toadjust to a specified density at x=0.

Transformed Probability Weights

The new probability weights obtained by applying thecomputer-implemented system and method, and computer-readable medium foruse with computer means, of the invention, to a set of cumulativeprobabilities. Also sometimes called Transformed Cumulative ProbabilityWeights, or Transformed Probabilities. Transformed probability weightsare produced in FIG. 1, step 106, and FIG. 6, step 106.

Tweaking Lambda

In the computer-implemented system and method, and computer-readablemedium for use with computer means, of the invention, any adjustment tothe lambda value, as applied to a given probability distribution, toproduce a tolerably close Wang Price to the currently known marketprice, for an underlying risk vehicle, as shown in FIG. 3, step 314, orFIG. 5, step 514. Tweaking lambda also can help produce a Wang Pricethat is tolerably close to the currently known market price for acontingent payoff, as shown in FIG. 4, step 415. Tweaking lambda is thesame as iterating lambda.

Two-Factor Model

In the two-factor version of the computer-implemented system and method,and computer-readable medium for use with computer means, of theinvention, one can adjust for risk, as well as adjust for parameteruncertainty, in the same process. While the first factor “lambda”remains to be the market price of risk, the second factor “k” iscalibrated by a Student-t distribution with k degrees of freedom. Thistwo-factor model, as shown in FIG. 6, applies a Student-t distributionin Step 605, rather than a normal distribution. Mathematically, thisnovel and valuable result is expressed as the following:F*(x)=Ψ[Φ⁻¹(F(x))+λ], where Ψ has a Student-t distribution.

The two-factor model is based here on the Student-t distribution in Step605 of FIG. 6, though more computationally intensive embodiments of theinvention allow for other distributions such as mixed normal orempirically constructed distributions. This two-factor model is animportant enhancement to the invention for pricing risks for very lowfrequency but very high severity exposures.

Underlying

A financial instrument whose behavior, or value, is referenced by aderivative financial instrument, for the purposes of determining thevalue of the derivative financial instrument.

Underlying Variable

A more technical name for the future value for an underlying riskvehicle. The underlying risk vehicle can be any asset, liability, or amix of assets and liabilities, traded or underwritten.

In more technical discussions of the uses, applications, and propertiesof the present invention, consider that, the underlying variable X isthe future value for a specified underlying risk vehicle. One purpose ofthe computer-implemented system and method, and computer-readable mediumfor use with computer means, of the invention, is to find the Wang Pricefor this underlying variable X, a useful data result, as shown in FIG.2, step 201.

Underlying Risk Vehicle

A universal definition for any group of one or more underwritten ortraded assets and liabilities, whose future risk-neutral price can beanticipated by applying the computer-implemented system and method, andcomputer-readable medium for use with computer means, of the invention,to a historically-known or computer-generated distribution of potentialoutcomes for that risk vehicle, comprised of a set of probabilities andattached cash values, along with a market price of risk for that riskvehicle.

The underlying risk vehicle can be a stock, bond, currency, commodity,or some other traded financial instrument, for example, IBM commonstock. The underlying risk vehicle can also be any collection ofinsurance, credit, health care, pension, or some other underwrittencontract obligations, for example, a catastrophe insurance claims index.The underlying risk vehicle can also be a customized contract of cashdelivery between two economic entities whose outcome probabilities andcash payoffs are already known, for example, a specified cash deliveryin the event that sustained wind speed during a specified period of timeexceeds a threshold.

The underlying risk vehicle is selected for computer-implemented systemand method, and computer-readable medium for use with computer means, ofthe invention, in FIG. 2, step 202; FIG. 3, step 302; and FIG. 5, step502.

Volatility

The standard deviation of the return for a financial instrument, orunderlying risk vehicle, over some period of time.

Wang Price

The expected future value, discounted for the risk-free interest rate,of the underlying risk vehicle, or, of any contingent payoff for thatunderlying. The Wang Price is determined by applying thecomputer-implemented system and method, and computer-readable medium foruse with computer means, of the invention, to a set of cumulativeprobabilities, to gain a new set of transformed probability weights, andthen by multiplying the cash values attached to those weights, to gain aset of weighted values. The total of these weighted values, whendiscounted by the risk-free interest rate, gives the Wang Price.

The Wang Price recovers the risk-neutral price that is calculated fromlognormal probability distributions by Black-Scholes and otherimplementations of modern options theory. The Wang Price also produces arisk-neutral price for all other probability distributions, which in theprior art have not been effectively priced.

The Wang Price is produced in FIG. 2, step 212, for an underlying riskvehicle identified as a group of one or more traded assets andliabilities; FIG. 3, step 312, for an underlying risk vehicle identifiedas a group of one or more traded assets and liabilities; FIG. 3, step319, for contingent payoffs for the outcome of an underlying riskvehicle identified as a group of one or more traded assets andliabilities; FIG. 5, 512, for the outcome of an underlying risk vehicleidentified as a group of one or more underlying assets and liabilities;and FIG. 5, step 519, for contingent payoffs for the outcome of anunderlying risk vehicle identified as a group of one or moreunderwritten assets and liabilities.

Wang Transform

A kernel, or core process, of the computer-implemented system andmethod, and computer-readable medium for use with computer means, thattransforms any cumulative probability to yield a new, distorted,cumulative probability. In more technical discussions of the uses,applications, and properties of the present invention, consider that,the Wang Transform kernel of the invention transforms the cumulativeprobability F(y) to yield a new cumulative probability F*(y), as shownin FIG. 1, steps 102 through 106. A two-factor model of the WangTransform kernel of the invention transforms the cumulative probabilityF(y) to yield a new cumulative probability F*(y), as shown in FIG. 6,steps 602 through 606.

Weather Derivative

A financial instrument that specifies a financial settlement betweencounterparties based on observed events of temperature, rainfall,snowfall, or wind speed.

Weighted Values

In the computer-implemented system and method, and computer-readablemedium for use with computer means, of the invention, the product ofmultiplying the cashflow values of a distribution to their newprobability weights, after decumulation, for an underlying risk vehiclefor a selected group of traded or underwritten assets and liabilities,as shown in FIG. 2, step 209; FIG. 3, step 309; FIG. 4, step 410; andFIG. 5, step 509, or, alternatively, for a contingent payoff, as shownin FIG. 3, step 317, FIG. 4, step 411, and FIG. 5, step 517.

SUMMARY OF THE INVENTION

The invention is a computer-implemented system and method, andcomputer-readable medium for use with computer means, that enablesportfolio managers to effectively price any traded or underwritten riskin finance and insurance with any historically-known orcomputer-generated probability distribution. The invention provides auniversal approach to pricing assets and liabilities, whether traded onan exchange or over-the-counter market, or underwritten for directrisk-transfer, whether grouped or segregated, or even positive ornegative in value.

It is an object and advantage of the invention to be applicable to theentire universe of probability distributions, not just normal orlognormal distributions. The prior art heavily relied on the normal orlognormal assumptions, and took ad hoc corrective measures in practiceto reflect departures from these assumptions. For instance, when usingBlack-Scholes formula to price options written on a stock, differentvolatility values are used for different strike prices. This correctivemeasure is inconsistent with the Black-Scholes assumption that stockprices follow a lognormal distribution and thus the same volatilityshould be used for all strike prices. In the special case of lognormallydistributed asset prices, this invention recovers the CAPM price for theexpected rate of return and the Black-Scholes price for options.

It is an object and advantage of the invention to allow risk managementprofessionals to use any probability distribution, as suggested byhistorical data or by forecasting models in their discretionaryactivities of pricing, underwriting, reserving, capitalizing,transferring, or trading various risk vehicles.

It is an object and advantage of the invention to price any blending ofassets and liabilities, whose net cash value can potentially take bothpositive and negative values. Most financial price models deal withstrictly assets, which cannot be negative in value. For instance, theminimum value of one share of common stock is zero, and cannot benegative. On the other hand, actuarial pricing methods deal solely withliabilities, or losses, which can only be negative in value.

With the emergence of integrated financial and insurance products, andwith insurance risks being traded in the capital market, the prior artis deficient to price such products. By allowing an underlying riskvehicle to take on positive or negative values, this invention providesconsistent pricing, where an asset can be treated as a negativeliability, or alternatively, a liability can be treated as negativeasset.

It is an object and advantage of the invention to be equally applicablefor both traded assets and underwritten risks. Traded assets are oftenpriced in relation to other traded assets, and a subjective probabilitymeasure (called Q-measure) can be implied from the market prices ofassets. For instance, the default probability of a bond can be inferredfrom its market interest-rate spreads over the risk-free treasury bonds.Underwritten risks, on the other hand, are evaluated using theirobjective probabilities (P-measure) of loss. In the prior art, there wasa deep gap between these two approaches. This need was reflected in thefrustrations in the communications between a capital marketsprofessional (who uses Q-measure in the course of trading) and aninsurance professional (who uses P-measure in the course ofunderwriting). This invention provides a pricing bridge between theQ-measure world and P-measure world. With this invention, a P-measurecan be easily converted to a Q-measure, and vice versa.

It is an object and advantage of the invention, after a series ofdata-processing steps, to produce a useful data result, called the WangPrice. The Wang Price is a fair valuation for the future price of a riskvehicle, where the mean of future expected outcomes, as weighted bytheir respective individual probabilities, has been adjusted for risk.

It is an object and advantage of this invention that when the inventionis applied to a traded financial instrument whose outcomes are normallydistributed, and the lambda value of the invention further iterated sothat the discounted Wang Price converges to equal the last market quotefor that financial instrument, the lambda value, as obtained by theprocess of this invention, is equal to the Sharpe Ratio.

It is an object and advantage of the present invention to generate alambda value, representing the market price of risk, for underlyingassets and liabilities when their current market prices and past pricemovements are known, or future market prices and price movements can beprojected, by a normal, or non-normal distribution of outcomes.

It is an object and advantage of the present invention to measure theuncertainty of a variable representing a future cashflow outcome, withinany historically-known or computer-generated distribution of outcomes.

It is an object and advantage of the present invention to assess therelative risk-adjusted values of traded portfolios, or the relativerisk-adjusted values of underwriting selections of managed accounts.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a flow chart of the preferred embodiment of the presentinvention where the invention may be applied to a cumulative probabilitydistribution of cashflows.

FIG. 2 is a flow chart showing how to price the future value of assetsand liabilities after selecting a known lambda value for their marketprice of risk.

FIG. 3 is a flow chart showing how to price a contingent payoff on anunderlying traded asset/risk.

FIG. 4 is a flow chart showing how to estimate lambda, or the marketprice of risk, for a group of assets and liabilities based on theirknown market prices.

FIG. 5 is a flow chart showing how to price a contingent payoff of anunderwritten risk.

FIG. 6 is a flow chart of the preferred embodiment of the presentinvention where a two-factor model may be applied to a cumulativeprobability distribution of cashflows.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

The foregoing and other objects, aspects, and advantages will be betterunderstood from the following detailed description of a preferredembodiment of the invention with reference to the drawings. As definedherein, the “method” refers to the data processing-system, acomputer-implemented method, and computer-readable medium of theinvention that prices financial or insurance risks.

The core process, or kernel, of the invention, is the Wang Transform.Every process of the invention requires the operation of the kernel, totransform every cumulative probability paired to a set of cashflowoutcomes, to yield a new, distorted, cumulative probability.Specifically, the cumulative probability F(y) is transformed to yield anew useful data result, the cumulative probability F*(y), as shown inFIG. 1, steps 102 through 106.

Assets and liabilities, whether traded or underwritten, are riskvehicles, which means that they are the legal contrivances forundertaking some type of capitalized risk—for example a financial orinsurance risk. One begins using the invention, by identifying anunderlying risk vehicle as a container of sorts, itself holding, a groupof one or more individual assets or liabilities, each of which can beeither traded or underwritten. The invention is capable of outputtingthe fair value of this underlying risk vehicle, which is defined as itsprice in a transactions, after an adjustment for risk that is manifestedin the tabled distribution of outcomes for that underlying risk vehicle.

If the underlying risk vehicle is one whose “market price of risk” isknown, or already inferred, then one would use FIG. 2 as the process forobtaining the fair value of that underlying risk vehicle, when that fairprice is not yet known, or, if the latest market price for theunderlying is not known. This fair value of the underlying risk vehicleis called the Wang Price. For one skilled in the art, the “market priceof risk” is the Sharpe Ratio for normally distributed outcomes, or,interchangeably, for outcomes with lognormal returns. This “market priceof risk” is also identified in the prior art literature as the lambdavalue.

If the risk vehicle is a contingent payoff, like an option, referencinga traded underlying risk vehicle, then one would use FIG. 3 as theprocess for obtaining the fair value of the contingent payoff inquestion. The fair value of the traded underlying risk vehicle isobtained by iterating lambda, whose starter value can be the tradingequivalent of the “market price of risk,” called the Sharpe Ratio, untilthe sum of weighted outcomes reflecting the variability of such fairvalue converges to equal the last known market price for the instrument.After this convergence has been accomplished, the function for thecontingent payoff is applied to each of the variable outcomes of theunderlying, to obtain the fair value of the contingent payoff. The fairvalue of the underlying risk vehicle obtained by this process isidentified as the Wang Price for the underlying risk vehicle. The fairvalue of the contingent payoff obtained by this process is identified asthe Wang Price for the contingent payoff.

If the last known market price for a traded underlying risk vehicle isknown, one would use FIG. 4 as the process for obtaining the true“market price of risk,” or lambda, for that risk vehicle. If the tradedunderlying risk vehicle has normally distributed outcomes, or,interchangeably, outcomes with lognormal returns, this process willproduce a “market price of risk” that is equal to the Sharpe Ratio. Ifthe traded risk vehicle, however, has non-normally distributed outcomes,or non-lognormal returns, this process will produce a “market price ofrisk” that is more accurate than the Sharpe Ratio.

If the risk vehicle is a contingent payoff, like a loss layer ofreinsurance, referencing an underwritten underlying risk vehicle, thenone would use FIG. 5 as the process for obtaining the fair value of thecontingent payoff in question. The fair value of the underwrittenunderlying risk vehicle is obtained by iterating lambda, whose startervalue can be the underwriting equivalent of the “market price of risk”called the “underwriting load of risk,” until the sum of weightedoutcomes reflecting the variability of such fair value converges toequal similar capital loadings for similarly underwritten risks. Afterthis convergence has been accomplished, the function for the contingentpayoff is applied to each of the variable outcomes of the underlying, toobtain the fair value of the contingent payoff. The fair value of theunderlying risk vehicle obtained by this process is identified as theWang Price for the underlying risk vehicle. The fair value of thecontingent payoff obtained by this process is identified as the WangPrice for the contingent payoff.

For greater precision in evaluating fair value of an underlying riskvehicle that is a group of one or more traded and underwritten assetsand liabilities whose data collections of sampled outcomes and sampledprobabilities may be incomplete, or for greater consideration inevaluating risk vehicles whose outcomes are rare, extreme, or outlying,then one would use FIG. 6 as the core process, or kernel, fortransforming the weighted probabilities of the risk vehicle in question.The core process in FIG. 6 is a two-factor model for evaluating fairvalue. The first factor is the “market price of risk” or itsunderwriting equivalent, and the second factor is called “parameteruncertainty,” to incorporate the possible inadequacy for sampledoutcomes and sampled probabilities used. The two-factor model of thecore process transforms the cumulative probability F(y) to yield a newcumulative probability F*(y), as shown in FIG. 6, steps 602 through 606.

Example 1 Finding the Fair Value for a Traded Risk Vehicle

For the first example of this method, refer to Table 1, which is anunsorted Intel stock price distribution for closing prices for 24monthly periods, from June 1998 until June 2000. Intel is identified asan underlying a risk vehicle, consisting of a group of only one tradedasset, so the method of the invention turns to FIG. 2 to find the fairvalue for that underlying risk vehicle at some point in the future. Thisfair value is called pricing, with adjustment for risk. Under theprocess of this invention, this pricing is obtained by generating auseful data results output, called the Wang Price.

In FIG. 2, the method starts by determining the objective of theprocess. The objective of the process is to find the Wang Price for thefuture value of the risk vehicle in question, 201. The future value ofIntel at 1 month from the time of the last market quote is theobjective. This satisfies steps 201, where the particular future datehas been described, by a horizon date of 1 month, and 202, where theselection of the group of one or many underwritten and traded assets andliabilities have been described, by Intel stock. Thus the underlyingrisk vehicle consists solely of Intel stock.

The method next selects a table of future prospective outcomes, whoseprojected cashflow values have assigned probabilities. The selectedtable of prospective future cashflow outcomes and respectiveprobabilities is found in Table 1.

Table 1 is an unsorted Intel stock price distribution. This table isgenerated, by one skilled in the art, by, first, listing the monthlycalendar periods of the monthly closes, in Column 1A. For example, themost recent monthly calendar period on this table is June 2000. Eachmonthly closing stock price, has been divided by the previous monthlyclosing stock price, to get a monthly return, in Column 1B. For example,the most recent monthly return on this table is 0.03084, for June 2000.

The current price, is the monthly closing stock price at the end of June2000, which is $133.688, in Column 1C. This current price serves as thebase price for future projected outcomes. By multiplying the currentprice to each monthly return, that is, by multiplying 1B to 1C, across24 rows, the table provides a table of prospective future cashflowoutcomes, in Column 1D. For example, the June 2000 monthly return of0.03084, is multiplied by the most recent stock price of 133.688, to get4.123. This result is then added to the most recent stock price of133.688, to get 137.811.

There are 24 future outcomes generated in this manner, each outcomehaving a 1/24 probability of taking place. For example, the futureoutcome generated by the June monthly return, as multiplied to thecurrent price, has a probability of 0.04167, which is 1/24.

With this pairing of prospective future cashflow outcomes, in Column 1D,with their respective probabilities, in Column 1E, the table satisfiesstep 203.

The next step in the method, 204, sorts the table of prospective futureoutcomes, by ascending cashflow value. The entire table of outcomes iscomprised of two columns, a column holding prospective future cashflowvalues, and a column holding their respective probabilities. Thesefuture cashflow values, and their respective probabilities, are sortedtogether, by ascending cashflow values, from lowest value to highestvalue, starting from the lowest cashflow value, and ending with thehighest cashflow value, as shown in Table 2.

Table 2 is the sorted Intel stock price distribution. Column 2A showsthe monthly closes that generated the prospective future cashflowvalues, after sorting. For example, the topmost monthly calendar returnon this table is May 1999. This is because the May 1999 monthly return,after being multiplied to the current price, created a product that wasthen added to the current price, generating the lowest prospectivefuture cashflow value, $109.44 in Table 1, Column 1D. When Table 1 wassorted to produce Table 2, this value was sorted to the top of Column2B.

Column 2C shows the respective probabilities associated with eachprospective future cashflow result. For example, the prospective futurecashflow result of $109.44 has a probability of 1/24, or 0.04167. Thesorted columns of 2B and 2C satisfy step 204 in the process.

The next step is to cumulate the sorted probabilities, so that the lastprobability equals the number 1. Cumulating means adding all of thevalues preceding, to the values at hand, thereby producing a cumulativeresult. For example, the cumulative probability of the $109.44 futureresult, is equal to the values of the preceding probabilities above it,plus the value of the probability at hand. The preceding probabilitiesfor $109.44 are 0, because there are no preceding probabilities in rowsabove $109.44 in the list. This value of 0 is added to the probabilityat hand, which is 0.04167, as shown at the top of Column 2D.

For the next prospective future value, $112.59, the cumulativeprobability is adding all of the probabilities preceding the respectiveprobability, in rows above, which is 0.04167, and adding the probabilityat hand, in the row across, which is 0.04167, to make 0.08333.

For the next prospective future value, $119.80, the cumulativeprobability is adding all of the probabilities preceding the respectiveprobability, in rows above, which is 0.08333, and adding the probabilityat hand, in the row across, which is 0.04167, to make 0.12500.

The entire column of 2D is cumulated in this fashion, until the lastprospective future value, $172.38, has its probability cumulated, byadding all of the preceding probabilities, to the probability at hand.All of the preceding probabilities, 0.95833, is added to the probabilityat hand, 0.04167, to get the number 1. This satisfies step 205.

With the cumulated probabilities calculated for each prospective futurecashflow result, the method introduces a value for the “market price ofrisk,” also called lambda, step 206.

For the purposes of this example, the “market price of risk” is theSharpe Ratio, which is the average return for each of the 24 months,minus the average risk-free rate for each of the 24 months, all dividedby the average standard deviation of the returns for each of the 24months. One skilled in the art is able to compute these values from themonthly closing returns in Table 1, to get 0.06281 for the averagereturn for each of the 24 months, and obtain an average risk-free rateof 0.005833 for each of the 24 months. The average standard deviation ofthe returns for each of the 24 months is 0.12591. The Sharpe Ratio, iscalculated as (0.06281-0.005833)/0.12591 to equal a “market price ofrisk” of 0.4525, under the assumption of lognormal returns for the 24months in the table. This step satisfies step 206.

The method on FIG. 2 moves with step 207 to apply the core process, orkernel, of the Wang Transform, as found on FIG. 1, to create newprobability weights, step 101. The core process starts by taking thecumulated probabilities of a distribution, as found in Table 2, Column2D. This satisfies step 102.

The method on FIG. 1 then applies an inversion of the standard normaldistribution to all of the cumulated probabilities of the distribution,103. The inversion of a standard normal distribution can be defined andgenerated in many computer programming languages, but for the purposesof simplicity in this preferred embodiment, the method applies the Excelfunction of NORMINV to each of the cumulated probabilities, followed bythe parameters 0,1.

The NORMINV function in Excel returns the inversion of the standardnormal cumulative distribution, for the specified probability weight,mean, and standard deviation, when populated by the following threeparameter values. X is the probability value corresponding to the normaldistribution, between the numbers 0 and 1 inclusive. The number 0 is thearithmetic mean of the distribution. The number 1 is the standarddeviation of the distribution. (NORMSINV is a summary function for theNORMINV function having a 0 mean and a 1 standard deviation, and can beused for the purposes described herein as well.)

For example, in Table 3, NORMINV of the first cumulated probability of0.04167 is −1.7317, as expressed in NORMINV(0.04167,0,1). NORMINV of thesecond cumulated probability of 0.08333 is −1.3830, and NORMINV of thethird cumulated probability of 0.12500 is −1.1503. The NORMINV of thelast cumulated probability of 1 is a generated positive infinity value,for which one skilled in the art substitutes an arbitrarily large finitenumber, of 500000. This column of individual inversely mapped results,in Column 3A, satisfies step 103.

The method on FIG. 1 then follows step 104, by taking the inverselymapped results of step 103, and shifting them, by the selected lambdavalue of step 206. Lambda represents the “market price of risk” and wasearlier calculated by the method, at step 206, by means of a SharpeRatio, to obtain 0.4525. This value of 0.4525 is added to the inverselymapped results in Column 3A, to get shifted results, in Column 3B. Forexample, the inversely mapped result of −1.7317 is shifted by the lambdavalue of 0.4525, to get the shifted result of −1.2791. The inverselymapped result of −1.3830 is shifted by the lambda value of 0.4525 to get−0.9305. The inversely mapped result of −1.1503 is shifted by the lambdavalue of 0.4525 to get −0.6978. This satisfies step 104.

The method on FIG. 1 then follows step 105, by applying the normaldistribution to each of these shifted results. For one skilled in theart, the standard normal distribution function is easily defined andgenerated in many generic computer programming languages, but for thepurposes of simplicity in this preferred embodiment, the method appliesthe Excel function of NORMDIST to each of the shifted results, followedby the parameters 0, 1, 1.

The NORMDIST function in Excel returns the standard normal cumulativedistribution, for the specified probability weight, mean, and standarddeviation, when populated by the following four parameters. X is thevalue for which one wants the distribution. The number 0 is thearithmetic mean of the distribution. The number 1 is the standarddeviation of the distribution. The number 1 is the logical value for acumulative value. (NORMSDIST is a summary function for the NORMDISTfunction having a 0 mean and a 1 standard deviation, and can be used forthe purposes described herein as well.)

For example, in Table 3, NORMDIST of the first shifted result of −1.2791is 0.10043, as contained in the expression, (NORMDIST(−1.2791),0,1,1).NORMDIST of the second shifted result of −0.9305 is 0.17607. NORMDIST ofthe third shifted result of −0.6978 is 0.24265. NORMDIST of the lastcumulated probability of 500000.4525 is a regenerated value of 1.

This column of transformed cumulative probability weights, in Column 3C,satisfies step 105. The cumulative probability weights have beentransformed by the core process of the Wang Transform, step 106,completing the core process, or kernel, of FIG. 1.

The method moves away from the completed core process, or kernel, of theWang Transform in FIG. 1, and back to FIG. 2, to decumulate thetransformed probability weights, step 208. Decumulating meanssubtracting the single value preceding, from the value at hand, therebyproducing a decumulated result.

In Table 4, the transformed probability weights of Column 3C, aredecumulated by subtracting the single weight one row above a particularweight, from that particular weight, to get a decumulated weight. Forexample, the transformed probability weight of 0.10043 has 0 subtractedfrom itself, because there is no probability weight one row aboveitself.

The transformed probability weight of 0.17607 has 0.10043 subtractedfrom itself, to make 0.07564, because 0.10043 is the probability weightone row above itself.

The transformed probability weight of 0.24264 has 0.17606 subtractedfrom itself, to make 0.06658, because 0.17606 is the probability weightone row above itself.

Step 208 consists of the continuing process of decumulating thetransformed cumulative probability weights, so that the firstdecumulated weight equals the first cumulative weight, the seconddecumulated weight equals the second cumulative weight minus the firstcumulative weight, the third decumulated weight equals the thirdcumulative weight minus the second cumulative weight, and so on,continuing until the last decumulated weight equals the last cumulativeweight minus the next-to-last cumulative weight.

At the bottom of Column 3C, the transformed probability weight of1.00000 has 0.98552 subtracted from itself, to make 0.01448, because0.98552 is the probability weight one row above itself. This satisfiesstep 208.

In Table 4, the results of decumulation are in Column 4A, as decumulatedprobability weights. These results reflect the distorted probabilityweights produced by the core process, or kernel, of the Wang Transform.One may compare these distorted probability weights, of Column 4A, withthe original probability weights, of Column 1E, as found in Table 1. Thehighest probability of Column 4A, 0.10042, is much higher than thehighest probability of Column 1E, 0.04167. The lowest probability ofColumn 4A, 0.01448, is much lower than the lowest probability of Column1E, 0.04167.

These distorted probability weights are highest for the worstprospective cashflow outcomes, and lowest for the best prospectivecashflow outcomes. In Table 4, one may note the distorted probabilityweight for the worst prospective cashflow of 109.44, at 0.10042, whichis the highest distorted probability weight. One may also note thedistorted probability weight for the best prospective cashflow of172.38, at 0.01448, which is the lowest distorted probability weight.

These decumulated probability weights, interchangeably called distortedprobability weights, add up to the number 1. This means that theyreflect the probabilities of a new distribution. This new probabilitydistribution is called a distorted probability weighted distribution.

The method then moves to step 209, by multiplying all of the prospectivefuture cashflow values to their distorted probability weights. In Table4, for example, the future prospective stock price of 109.442 in Column3B is multiplied by the distorted probability weight of 0.10042 in 4A,to get a weighted value of 10.990. The future prospective stock price of112.442 is multiplied by the distorted probability weight of 0.07564, toget a weighted value of 8.516. The future prospective stock price of119.798 is multiplied by the distorted probability weight of 0.06658, toget a weighted value of 7.976. This satisfies step 209.

The method then moves to step 210, which sums all of the weighted valuesto get an undiscounted Wang Price. In Table 4, all 24 of the weightedvalues in Column 4B are added together, to equal $134.728. This is theundiscounted Wang Price for Intel stock in 1 month. This satisfies step210.

The method finally discounts the Wang Price by the risk-free interestrate in step 211. The undiscounted Wang Price in 1 month is $134.728. Bymultiplying the risk-free interest rate of 1 month to this Wang Price,and subtracting this amount from the Wang Price, the method obtains thediscounted Wang Price. The risk-free interest rate of 1 month is0.005833, as obtained from the calculation of the Sharpe Ratio in step206. Multiplying the Wang Price of $134.728 by 0.005833, the method gets$0.785, which is further subtracted from $134.728 to obtain a discountedWang Price of $133.944, as shown on the bottom of Column 4B. Thissatisfies step 211, and obtains the discounted Wang Price as processedthrough FIG. 2, step 212. This completes FIG. 2. This completes Example1.

The method of the process for FIG. 2 can be used to obtain a price,after adjustment for risk, for an underlying risk vehicle that consistsof a group of any number of other traded assets and liabilities, such asstocks or other equity securities, bills, bonds, notes, or other debtsecurities, currencies of various countries, commodities of physical,agricultural, or financial delivery, asset-backed or liability linkedsecurities or contractual obligations, and weather derivatives and otherobservable physical phenomena whose outcomes can be linked to financialoutcomes. As generated by the method, this price, is called the WangPrice.

The Wang Price, after discounting, is a useful data result, because itrepresents the present fair value of an underlying risk vehicle thatitself can be an asset or a liability or a group of any greater numberof assets or liabilities. This present fair value can be compared to thepresent fair value of other underlying risk vehicles or of otherfinancial instruments, on an even playing field, so that risk managementprofessionals can identify, monitor, acquire, and dispose of assets andliabilities according to relative comparisons of expected portfoliorisks and returns.

Example 2 Finding the Fair Value for an Option on a Traded UnderlyingRisk Vehicle

For the second example of this method, refer to Table 5, which is aEuropean call option at a strike price of $140 on an Intel stock pricedistribution for closing prices for 24 monthly periods, from June 1998until June 2000, whose probabilities have already been transformed bythe previous example. Intel has been identified already as a tradedunderlying risk vehicle, so the method of the invention turns to FIG. 3to find the fair value for the contingent payoff on that underlying riskvehicle at some point in the future. Under the process of thisinvention, this pricing is obtained by generating a useful data resultfor the underlying stock, in the form of an output called the WangPrice, and then applying a payoff function of MAX(140−X,0) to thedistorted probabilities of the underlying stock, representing X.

In FIG. 3, the method starts by determining the objective of theprocess. The objective of the process is to find the Wang Price for thefuture value of the contingent payoff on the underlying risk vehicles,301.

The future value of the European call option whose strike price is $140at 1 month from the time of the last market quote is the objective. Thissatisfies steps 301, where the particular future date has beendescribed, by a horizon date of 1 month, and 302, where the selection ofa traded underlying risk vehicle has been described as consisting of asingle asset namely Intel stock.

One skilled in the art would notice that the steps 303 through 312exactly replicate the steps 203 through 212 from Example 1, where theWang Price for Intel stock, at $133.944, was obtained at step 212. Inthis example, this same Wang Price for Intel stock, is obtained at step312.

At step 313, however, the method requires a decision. Is the Wang Pricefor the underlying close enough to the last quoted market price? Thelast quoted market price for Intel stock is $133.688, which is $0.25less than the discounted Wang Price of $133.944. Depending on thetolerance of one skilled in the art for relative lack of precision, themethod allows for application of a payoff function to each variableoutcome of the underlying, in step 315, or requires further iteration ofthe lambda value, step 314.

For the purposes of this example, the method requires further iterationof the lambda value. The provisional lambda value in Example 1 was0.4525, based on the Sharpe Ratio. The method tweaks this provisionallambda value, at step 314, until the generated Wang Price, from steps306-312, converges to equal the last market price of $133.688, step 313.After trial and error, the calibrated lambda value of 0.4685 produces aWang Price of $133.688. This satisfies step 313.

One skilled in the art may notice that the calibrated lambda value of0.4685 is different from the Sharpe Ratio lambda of 0.4525. Thecalibrated lambda adjusts for the fact that the returns for theunderlying Intel stock, as listed in Table 1, Column 1B, are not trulylognormal. If the returns for the underlying Intel stock were trulylognormal, the Sharpe Ratio value for provisional lambda would produce adiscounted Wang Price equal to the last market price.

A calibrated change in lambda value produces calibrated changes in Table5, as shown in the cumulated probabilities of Column 2D, transformedprobabilities of Column 5A, and distorted probability weights of Column5B, when compared to those in Example 1.

With the calibrated lambda of 0.4685, the method has already completedthe steps to finding a discounted Wang Price for the underlying Intelstock, as found in the bottom of Column 5C.

The method then moves to application of the payoff function to theprospective future cashflow outcomes of the underlying, step 315. InTable 5, the payoff function of MAX(140−X,0) is applied to theprospective future cashflow outcomes of Column 2B, as X, with alloutputs generated in Column 5D.

For example, the prospective future cashflow outcome of $109.442 minusthe $140 call price, is a negative number, so the MAX(140−X,0) payofffunction generates a 0 value.

The prospective future cashflow outcome at the bottom of Column 2B, of$172.379, minus the $140 call price, is a positive number of $32.38, sothe MAX(140−X,0) payoff function generates a $32.38 value. With alloutputs generated in Column 5D, this satisfies step 315.

The method then moves to step 316, multiplying the payoff values totheir distorted probability weights. In Table 5, the payoff values arefound in Column 5D, and the distorted probability weights are found inColumn 5B. When they are multiplied together, they generate outputs inColumn 5E.

For example, the contingent payoff value of $0.00 at the top of Column5D, can be multiplied to the distorted probability weight 0.1033 at thetop of Column 5B, to equal a weighted value of $0.00 at the top ofColumn 5E.

At the bottom of the respective columns, however, the contingent payoffvalue of $32.38 is multiplied to the distorted probability weight of0.0139, to equal a weighted value of $0.450. With all outputs generatedin Column 5E, this satisfies step 316.

The method now moves to step 317, where the weighted values for thecontingent payoff are added together, to obtain a Wang Price. In Table5, the weighted values of Column 5E are added together, to equal $4.537,which is the undiscounted Wang Price for the option in 1 month. Thissatisfies step 317.

The method is completed by discounting the Wang Price for the option bythe risk-free interest rate, step 318. The risk-free interest rate of 1month is 0.005833, as obtained from the calculation of the Sharpe Ratioin step 206. Multiplying the Wang Price of $4.537 by 0.005833, themethod gets $0.027, which is further subtracted from $4.537 to obtain adiscounted Wang Price of $4.510, as shown on the very bottom of Column5E. This satisfies step 318, and obtains the discounted Wang Price asprocessed through FIG. 3, step 319. This completes FIG. 3. Thiscompletes Example 2.

One skilled in the art may notice that the discounted Wang Price for theoption is $4.510, and the discounted Black-Scholes price for the sameoption is $4.171. The Wang Price adjusts for the fact that the returnsfor the underlying Intel stock, as listed in Table 1, Column 1B, are nottruly lognormal. If the returns for the underlying Intel stock weretruly lognormal, the Sharpe Ratio value for provisional lambda wouldproduce a discounted Wang Price, equal to the Black-Scholes price.

The invention calibrates an accurate option price for an underlyingfinancial instrument, regardless of whether that instrument has a normalor non-normal set of prospective future cashflow outcomes. Thiscalibration of an accurate option price is a useful data result of theinvention.

The method of the process for FIG. 3 can be used to obtain a price,after adjustment for risk, for contingent payoffs on any other tradedunderlying risk vehicle, consisting of any group of one or more assetsor liabilities, such as options on stocks or other equity securities,options on bills, bonds, notes, or other debt securities, options oncurrencies of various countries, options on commodities of physical,agricultural, or financial delivery, options on asset-backed orliability linked securities or contractual obligations, and options onweather derivatives and other observable physical phenomena whoseoutcomes can be linked to financial outcomes.

The Wang Price, after discounting, is a useful data result, or output,because it represents the present fair value of an underlying riskvehicle for any grouping of one or more assets or liabilities. Thispresent fair value can be compared to the present fair value of otherunderlying risk vehicles, on an even playing field, so that riskmanagement professionals can identify, monitor, acquire, and dispose ofunderlying risk vehicles according to expected portfolio risks andreturns.

Example 3 Finding the Market Price of Risk for a Bond Subject to RatingMigration

For the third example of this method, refer to Table 8, which provides aseries of outcomes for a BBB-rated corporate bond, whose coupon rate isat 6%, with the risk-free interest rate at 5%. The rating for the BBBbond may migrate over one year, to AAA, at best, or to Default, atworst, as shown in Column 8A. Regardless of the future rating, theprospective future cashflows for the coupon rate on this bond remains at6%, unless the bond falls into default, as shown in Column 8B. Theprospective forward prices for the bond in one year, given theprospective change in rating, is shown in Column 8C. With the addedvalue of coupon payment, at 6% of the $100 book value, the total forwardvalues of the bond are shown in Column 8D. The various probabilitiesattached to the migrations are shown in Column 8E.

The corporate bond is a traded underlying risk vehicle, so the method ofthe invention turns to FIG. 4 to find the “market price of risk” for theunderlying risk vehicle—consisting of a single asset, namely thecorporate bond—at some point in the future. This “market price of risk”is a useful data result, because it can be compared favorably, orunfavorably, to the “market price of risk” of other underlying riskvehicles, having otherwise similar expected returns. Under the processof this invention, iterating the “market price of risk” is used todiscount the future Wang Price, until the discounted Wang Price equalsthe last market price for that underlying risk vehicle.

In FIG. 4, the method starts by determining the objective of theprocess. The objective of the process is to find the “market price ofrisk” for the future value of the underlying risk vehicle in question,401. The future value of the corporate bond 1 year from the time of thelast market quote is the objective. This satisfies steps 401, where theparticular future date has been described, by a horizon date of 1 year,and 402, where the selection of a traded underlying risk vehicle hasbeen described, by the corporate bond.

The method next selects a table of future prospective outcomes, whoseprojected cashflow values have assigned probabilities, step 403. Theselected table of prospective cashflow future outcomes and respectiveprobabilities is found in Table 8, in Columns 8D and 8E. This satisfiesstep 403.

The method next sorts these prospective cashflow future outcomes, andrespective probabilities, in ascending order, from lowest to highest,step 404. These sorted outcomes, with their respective probabilitiesstill attached, are shown in Table 9, Columns 9A and 9B. This satisfiesstep 404.

The method next cumulates the sorted probabilities so that the lastprobability equals the number one, step 405. These cumulatedprobabilities are shown in Column 9C. The cumulated probability for thelast sorted outcome, 109.37, equals the number 1. This satisfies step405.

The method next selects a lambda value, as the “market price of risk.”This selection is a provisional value, which will be reselected by themethod again and again, by iteration, until the discounted Wang Priceconverges to equal the $100.00, the last market price for the bond.

One skilled in the art perceives that the returns in Table 9 are notlognormal, so that the Sharpe Ratio can only provide a provisional, orstarter, lambda value. For the purposes of this example, the SharpeRatio, is the average of weighted bond returns for the year, minus theaverage risk-free rate for the year, all divided by the average standarddeviation of the weighted bond returns for the year.

One skilled in the art is able to compute these values from theprospective future cashflow outcomes, and their respectiveprobabilities, to get 0.0709 for the average bond return for the year,and obtain an average risk-free rate of 0.0500 for the year. The averagestandard deviation of the bond returns for the year is 0.0299. TheSharpe Ratio, is calculated as (0.0709−0.0500)/0.0299 to equal a “marketprice of risk” of 0.6980, under the assumption of lognormal returns.Lambda thus equals 0.6980, for now. This satisfies step 406.

The method next applies the core process, or kernel, of the WangTransform, to the cumulated probabilities, step 407. The Wang Transformis found in FIG. 1, but one skilled in the art using Excel can summarizethe entire kernel with a line of combined code, as follows:COLUMN_(—)9D=NORMDIST(NORMINV(COLUMN_(—)9C,0,1)+LAMBDA,0,1,1))

The core process of the Wang Transform starts by taking the cumulatedprobabilities of a distribution, as found in Table 9, Column 9C. Column9C appears on the innermost parenthesis of the above equation. Thissatisfies step 102.

The method on FIG. 1 then applies an inversion of the standard normaldistribution to all of the cumulated probabilities of the distribution,103. The inversion of a standard normal distribution can be generated inmany computer programming languages, but we list the Excel function ofNORMINV, as applied to each of the cumulated probabilities. The NORMINVis applied to (COLUMN_(—)9C) followed by the parameters 0,1. Thissatisfies step 103.

The method on FIG. 1 then follows step 104, by taking the expression of(NORMINV(COLUMN_(—)9C,0,1) and applies a shift, by the selected lambdavalue of step 406. The value of 0.6980 is thus added to the(NORMINV(COLUMN_(—)9C,0.1)). This satisfies step 104.

The method on FIG. 1 then follows step 105, by applying the normaldistribution to each of these shifted results. We apply the Excelfunction of NORMDIST to the expression NORMINV(COLUMN_(—)9C,0,1)+LAMBDA,followed by the parameters 0,1,1 to get a complete kernel, or coreprocess, expression of:NORMDIST(NORMINV(COLUMN_(—)9C,0,1)+LAMBDA,0,11).

For example, in Table 9, the Wang Transform of the first cumulatedprobability of 0.0018, as shown in Column 9C, is 0.0134, as shown inColumn 9D. The Wang Transform of the last cumulated probability of1.0000, as shown in Column 9C, is 1.0000, as shown in Column 9D.

This column of transformed cumulative probability weights, in Column 3C,satisfies step 105. The cumulative probability weights have beentransformed by the core process of the Wang Transform, step 106,completing the core process, or kernel, of FIG. 1.

The method moves away from the completed core process, or kernel, of theWang Transform in FIG. 1, and back to FIG. 4, to decumulate thetransformed probability weights, step 408. In Table 9, the transformedprobability weights, as shown in Column 9D, are decumulated, as shown inColumn 9E. For example, the transformed probability weight at the top ofColumn 9D, 0.0134, is decumulated to 0.0134, at the top of Column 9E.But the transformed probability weight at the bottom of Column 9D,1.0000, is decumulated to 0.00001, as shown at the bottom of Column 9E.This satisfies step 408.

Notice that the new probability weights are distorted from theiroriginal probability weights, because of the effects of the kernel, orcore process, of the invention, the Wang Transform. These distortionsare due to our selection of the lambda value.

In FIG. 4, the method moves from decumulation to immediate applicationof the payoff function. This payoff function is the underlying bonditself. One skilled in the art understands that an underlying asset canbe viewed as a contingent payoff, with a resulting contingent cash valueamount identical to that of the underlying cash value.

The applied payoff function is in Table 9, in Column 9F, and isidentical to the prospective values of Column 9A. For example, the worstprospective future cashflow outcome is $51.13, at the top of Column 9A,which is also the worst prospective payoff function value, at the top ofColumn 9F. This satisfies step 409.

The payoff function values from Column 9F are now multiplied to theirnew probability weights, 9E, to provide results in Column 9G, step 410.For example, the top payoff function value in Column 9F, $51.13, ismultiplied to the top new probability weight in Column 9E, 0.01344, toequal the top weighted value in Column 9G, $0.687. This is the valuecontribution of the default outcome in 1 year, to the provisional WangPrice of the bond. One trained in the art understands that the $83.222value in Column 9G, represents the value contribution of the BBB-ratingoutcome in 1 year, to the provisional Wang Price of the bond. Thissatisfies step 410.

The method now adds all of these weighted values in Column 9G to obtaina provisional Wang Price for 1 year from now, step 411. This price is$105.38. This satisfies step 411.

The method then moves to discount the provisional Wang Price by therisk-free interest rate, 412. The risk-free interest rate is 0.0500 for1 year, and so the price of $105.38 is discounted to $100.37, which isthe present value of the provisional Wang Price. This satisfies step412, and provides the Wang Price for the contingent payoff of theunderlying, 413.

At step 414, however, the method requires a decision. Is the Wang Pricefor the contingent payoff close enough to the last quoted market pricefor that contingent payoff? The last quoted market price for the bondwas $100.00, which is $0.37 less than the discounted Wang Price of$100.37. Depending on the tolerance of one skilled in the art forrelative lack of precision, the method allows an end to the process, or,for further iteration of the lambda value, step 415.

For the purposes of this example, the method in FIG. 4 requires furtheriteration of the lambda value. The provisional lambda value in Table 9was 0.6980, based on the Sharpe Ratio. The method tweaks thisprovisional lambda value, at step 415, until the generated Wang Price,from steps 406-413, converges to equal the last market price of $100.00,step 414, before ending. After trial and error, the calibrated lambdavalue of 0.788 produces a discounted Wang Price of $100.00. The effectsof calibrating lambda to 0.788 is shown in Table 10, with each of thecolumns representing the steps of 406-412. This satisfies step 414, andcompletes FIG. 4, step 416. This completes Example 3.

The calibrated market price of risk for the bond, as generated by theinvention, is different than the Sharpe Ratio, because the weighteddistributions of prospective future cashflow outcomes does not exhibitlognormal returns. One skilled in the art is able to multiply Columns 8Dand Columns 8E, and adding the weighted values, to get a meanexpectation for the return of the bond, $107.28. All other things beingequal, a bond with a mean expected return of $107.28, but with a lowlambda value, like 0.3, is less uncertain, than a bond with the samereturn, but with a higher lambda, like 0.5.

The invention calibrates an accurate “market price of risk” for afinancial instrument, regardless of whether that instrument has a normalor non-normal set of prospective future cashflow outcomes. Thiscalibration of the accurate “market price of risk” is a useful dataresult, for the purposes of portfolio risk management.

With the calibration of an accurate “market price of risk,” the processof FIG. 4 can also be used to price the prospect of bond, loan, ormortgage default, or the prospect for a defaulted bond, loan, ormortgage to recover and resume payments. Fractionated probabilities forthe prospective future cashflow outcomes can be further refined, orsegregated, to reflect variations in the estimations of recovery fromdefault. The process of FIG. 4 can also be used to price thesecuritizations of credit card, mortgage, loan, or other accountreceivables.

With the calibration of an accurate “market price of risk,” the processof FIG. 4 can be used to obtain a price, after adjustment for risk, forcatastrophe bonds, where the scheduled coupon payments and principalpayments may be reduced due to a specified catastrophic event.

The process of FIG. 4 can also be used to price the prospect of default,or the prospect for a defaulted bond to recover and resume payments.Fractionated probabilities for the prospective future cashflow outcomescan be further refined, or segregated, to reflect variations in theestimations of recovery from default.

The output of the Wang Price, after discounting, is a useful dataresult, because it represents the present fair value of an underlyingrisk vehicle consisting of a group of one or more assets or liabilities.This present fair value of the underlying risk vehicle can be comparedto that of other underlying risk vehicles, on an even playing field, sothat risk management professionals can identify, monitor, acquire, anddispose of underlying risk vehicles according to expected portfoliorisks and returns.

Example 4 Finding the Fair Value for an Underwritten Catastrophe

For the fourth example of this method, refer to Table 11, which providesa series of outcomes for a Richter Scale earthquake event for somepopulated epicenter, in Column 11A, with payments contracted for thefirst of any such event over 1 year, according to degree of severity, inColumn 11B. Any payout will be paid at the end of the year. Prospectivefuture cashflow outcomes remain at $0.00, if all Richter Scale eventsremain below 6.00 for the year. The payouts begin with $100.00 for thefirst Richter Scale event of 6.00 or higher, with a capped payout of$271.83 for a first Richter Scale event of 7.00 or greater. The variousprobabilities attached to Richter Scale severity are shown in Column11C.

The earthquake contract is an underwritten risk with a contingentpayout, so the method of the invention turns to FIG. 5 to find the fairprice for the contract at some point in the future.

In FIG. 5, the method starts by determining the objective of theprocess. The objective of the process is to find the Wang Price for thefuture value of the underlying risk vehicle in question, 501. The fairvalue of the earthquake contingency contract in 1 year, from theperspective of a Small Insurance Company, is the objective. Such a fairvalue, in the insurance world, is called a pure premium. The purepremium would be charged by the Small Insurance Company, to break evenon the standalone cost of the contract, after an adjustment for risk.

This satisfies steps 501, where the particular future date has beendescribed, by a horizon date of 1 month, and 502, where the selection ofunderwritten liabilities has been described, by an earthquakecontingency contract, with payouts described in Table 11, as offered bySmall Insurance Company.

The method next selects a table of future prospective outcomes, whoseprojected cashflow values have assigned probabilities, step 503. Theselected table of prospective future cashflow outcomes and respectiveprobabilities is found in Table 11. The prospective future cashflowoutcomes are found in Column 11B, and their respective probabilities arefound in Column 11C. This satisfies step 503.

The method then moves to sort the entire table of outcomes by ascendingcashflow values, from lowest cashflow value to highest cashflow value,step 504. These sorted cashflows, and their respective probabilities,are shown in Table 12. For example, the lowest cashflow value, at thetop of Column 12A, is $0. This cashflow is paid out if there is noearthquake measuring at 6.0 or greater on the Richter Scale. The highestcashflow value, at the bottom of Column 12A, is $271.83. This cashflowis paid out for the first of any earthquakes measuring 7.0 or greater onthe Richter Scale. This satisfies step 504.

The method then moves to cumulate the sorted probabilities so that thelast probability equals the number 1, step 505. The cumulatedprobabilities are shown in Column 12C. This satisfies step 505.

The method now selects a lambda value, step 506, for the “underwritingload of risk.” This “underwriting load of risk” is the negatively signedversion of the “market price of risk.” The prospective future cashflowamounts are paid out, and not received in, because the earthquakecontingency contract is a liability. The fair value for the liabilitywill be the pure premium received in to cover these payouts, afteradjusting for risk.

For the purposes of this example, the method selects a lambda value of−0.3, to indicate the underwriting load of risk. This lambda value isselected by Small Insurance Company from available records of earthquakeexperience and prevailing market rates for pricing insurance. Thissatisfies step 506.

The method next applies the kernel, or core process, of the WangTransform to the probability weights, to create new probability weights,step 507. This kernel can be found in FIG. 1. The method on FIG. 1follows steps 101-106, by applying the expression of:NORMDIST(NORMINV(COLUMN_(—)12C,0,1)+LAMBDA,0,1,1)). The transformedprobability weights are shown in Column 12D. This satisfies all of thesteps of FIG. 1 in the Wang Transform kernel. This satisfies step 507.

The method then decumulates the transformed probability weights intodistorted probability weights, step 508. The decumulated probabilityweights are shown in Column 12E. This satisfies step 508.

The method next multiplies the prospective future cashflow values totheir new probability weights, step 509. The prospective future cashflowvalues are shown in Column 12A. The new probability weights are shown inColumn 12E. The weighted values of the payoffs are shown in Column 12F.This satisfies step 509.

The method then sums all of the weighted values to get an undiscountedWang Price, step 510. These weighted values equal $59.05, as shown atthe bottom of Column 12F. This price is the undiscounted Wang Price.This satisfies step 510.

The method next discounts the Wang Price by the risk-free interest rate,step 511. For the purposes of this example, the risk-free interest rateis 0.07, and so the discounted Wang Price equals $55.18. This is thefair value for the earthquake contingency contract. This satisfies step511, and completes the first part of the process of FIG. 5, at step 512.This completes Example 4.

One skilled in the art may calculate the difference between the meanexpected payout of the earthquake contingency contract, and the fairprice, after adjustment for risk, for that contract. The mean expectedpayout for the contract is found by multiplying the prospective futurecashflow outcomes, as found in Column 12A, by their respective originalprobability weights, as found in Column 12B, and then summing up theirweighted values. The mean expected payout for the contract is thus$35.77. This is almost $20 less than the fair value for the contract,after adjustment for risk.

The process of FIG. 5 can also be used to price underwritten liabilitiesin credit, insurance, and pensions, where a certain amount of breakevenmoney needs to be received by the underwriter assuming the liabilities.

The Wang Price, after discounting, is a useful data result, because itrepresents the present fair value of an asset or liability. This presentfair value can be compared to the present fair value of other financialinstruments, on an even playing field, so that risk managementprofessionals can identify, monitor, acquire, and dispose of underlyingrisk vehicles consisting of a group of one or more assets andliabilities according to expected portfolio risks and returns.

Example 5 Finding the Fair Value for a Reinsured Layer of anUnderwritten Catastrophe

For the fifth example of this method, refer to Table 13, which providesthe same series of prospective future cashflow outcomes, and respectiveprobabilities, as the last example.

The underwritten risk is the transfer of a specific layer of theearthquake contingency contract, from Small Insurance Company, to theVery Large Reinsurer, so that all contracted payouts in excess of $200are reinsured. This reinsured layer is a contingent payout for anunderwritten risk, with the function MAX(X−200,0), so the method of theinvention turns to the process in FIG. 5 to find the fair price for thereinsured layer.

In FIG. 5, the method starts by determining the objective of theprocess. The objective of the process is to find the Wang Price for thefuture value of the contingent payout in question, 501. The fair valueof the reinsured layer in 1 year, from the perspective of a SmallInsurance Company, is the objective. This satisfies step 501.

The process in FIG. 5 covers the Wang Price for the underlyingunderwritten contract, which is covered in steps 501 to 512. Theunderwritten underlying risk vehicle is the same earthquake contingencycontract that was priced in Example 4. The discounted Wang Price forthis contract was $55.18, as found in step 512 of the Example 4 process.

The method then moves from step 512 in this Example 5 process, to step513. At step 513, however, the method requires a decision. Is the WangPrice for the underlying close enough to the other pure premiums forsimilar underwritten liabilities? The last quoted market price forearthquake contingency contracts, for similar payouts, was indeed near$55.18. Depending on the tolerance of one skilled in the art forrelative lack of precision, the method allows for application of apayoff function to each variable outcome of the underlying, in step 515,or requires further iteration of the lambda value, step 514.

For the purposes of this example, this Wang Price is indeed closeenough, satisfying step 513.

The method next moves to step 515, by applying the payoff function tothe underlying underwritten liability. The payoff function isMAX(X−200,0) to the projected future cashflow outcomes of the earthquakecontingent contract. These outcomes are shown in Column 12A. The resultof applying the payoff function to these outcomes is shown in Column13A.

For example, the projected future cashflow outcome at the top of Column12A is $0. After applying the payoff function ofMAX(Column_(—)12A-200,0), the contingent payoff at the top of Column 13Ais $0.

The projected future cashflow outcome at the bottom of Column 12A,however, is $271.83. After applying the payoff function ofMAX(Column_(—)12A−200,0), the contingent payoff at the bottom of Column13A is $71.83.

With the payoff function completed in Column 13A, this satisfies step515.

The method now moves to multiply the contingent cashflow values to theirnew probability weights, step 516. In Table 13, the contingent cashflowvalues, as found in Column 13A, are multiplied to their respective newprobability weights, as found in Column 12E. One skilled in the artremembers that these probability weights were first generated in Example4, in step 508, as a result of applying lambda to the get the calibratedWang Price for the underlying underwritten earthquake contingencycontract.

As an example of step 516, the contingent payoff at the top of Column13A is $0. The distorted probability at the top of Column 12E is 0.7060.By way of multiplication, the resulting weighted payoff is $0, at thetop of Column 13B.

The contingent payoff at the bottom of Column 13A, however, is $71.83.The distorted probability at the bottom of Column 12E is 0.1194. By wayof multiplication, the resulting weighted payoff is $8.58, at the bottomof Column 13B.

With the weighted values completed in Column 13B, this satisfies step516.

The method then moves to sum the weighted values to get an undiscountedWang Price for the reinsured layer, step 517. In Table 13, the weightedvalues add up to $9.41, as found below the bottom of Column 13B. This isthe undiscounted Wang Price. This satisfies step 517.

The method now moves to discount this Wang Price by the risk-freeinterest rate, step 518. With a risk-free interest rate of 0.07, thediscounted Wang Price is $8.79. This is the fair value for the reinsuredlayer of the earthquake contingency contract, covering any payout forthat contract, that exceeds a payout of $200. Small Insurance Companywould pay the Very Large Reinsurer this amount of money, as a purepremium for taking on this layer of liability. This satisfies step 518,and completes the second part of the process of FIG. 5, at step 519.This completes Example 5.

One skilled in the art may notice that the discounted Wang Price for thereinsurance layer is $8.79, but that there is no discountedBlack-Scholes price considered for the reinsurance layer. To one skilledin the prior art, using a Black-Scholes model to price a slice of anon-normally distributed earthquake contract would not make sense. Theexample provides no time series of varying outcomes and returns, toproduce a volatility parameter. The authoritative example provides noway to execute a riskless hedge against the reinsured layers during theterm of the contract. A Black-Scholes model would not provide ameaningful basis for pricing such a reinsurance contract.

The invention calibrates an accurate option price for an underlyingfinancial instrument, regardless of whether that instrument has a normalor non-normal set of prospective future cashflow outcomes, andregardless of whether that instrument is an asset or liability, andregardless of whether that instrument is traded or underwritten. Thiscalibration of an accurate option price is a useful data result of theinvention.

The invention calibrates an accurate contingent price for an underlyingfinancial instrument, regardless of the structure of the contingentpayoff, option, or payoff function. This calibration of an accuratecontingent price, regardless of the structure of the contingency, is auseful data result of the invention.

As an alternative to this example, a reinsurance layer may be priced byexceedence probabilities. An exceedence probability is the overallprobability that the future value of an underlying risk vehicle willexceed a certain amount of money, in terms of either gains as an asset,or losses as a liability. Exceedence probabilities are used frequentlyin underwriting financial obligations in insurance, credit, health care,and pensions, and especially in catastrophe insurance pricing.

In Table 12, the exceedence probabilities for the earthquake contingencycontract are provided in a new Column 12C. One skilled in the artunderstands that exceedence probabilities may be generated bysubtracting the cumulations preceding and including the probabilityweights at hand from a probability of certainty, or 1. Columns 12A and12B show the results of the method for the previous example, steps501-504, but with exceedence probabilities in step 503 in place of thecumulated probabilities. This is because exceedence probabilities arethemselves generated cumulations.

For example, the exceedence probability for the top probability weightin Column 12B is 0.8000. The cumulations preceding and including theprobability weights at hand is also 0.8000. The exceedence probabilityfor this layer, is therefore 1 minus 0.8000, or 0.2000, as shown at thetop of New Column 12C.

The exceedence probability for the next probability weight in Column 12Bis 0.02000. The cumulations preceding and including the probabilityweights at hand is 0.8000 plus 0.02000, or 0.82000. The exceedenceprobability for this later, is therefore 1 minus 0.82000, or 0.18000, asshown at the top of New Column 12C.

These exceedence probabilities are the results of a cumulation, step505, and a lambda value of −0.3 selected, step 506. The method movesnext to transform these cumulated exceedence probabilities, under thekernel, or core process, of the Wang Transform, as shown in FIG. 1,steps 101-106. After decumulation, FIG. 5, step 508, the distortedprobability weights are shown in Column 12E. For example, the topdistorted probability weight of column 12E is 0.7060. This satisfiessteps 506-508.

The original cashflow values in Column 12A can be multiplied by theirnew respective probability weights, step 509. For example, the payoutamount at the top of Column 12A is $0, and the distorted probability atthe top of Column 12E is 0.7060. The weighted payoff at the top ofColumn 12F is $0.00.

But the payout amount at the bottom of Column 12A is $271.83, and thedistorted probability at the bottom of Column 12E is 0.1194. Theweighted payoff at the bottom of Column 12F is therefore $32.47. This$32.47 is the undiscounted price of the cost of a probability layer in areinsurance contract. The probability layer that is priced is theprobability that an earthquake will be at least 7.0 on the RichterScale. One skilled in the art can discount the $32.47 by the risk-freeinterest rate, to obtain the fair value for assuming this probabilitylayer of severe outcome. This probability layer is called an excess ofseverity layer, because it a layer that is paid only when the severityof an earthquake is in excess of 7.0 on the Richter Scale.

The method of the process for FIG. 5 can be used to obtain a price,after adjustment for risk, for contingent payouts of underwritten assetsand liabilities, such as stop-loss layers, probability layers, excess ofseverity layers, or excess of loss layers, in reinsurance, or to priceoptions, contingencies, slices, or provisions in underwrittenliabilities in credit, insurance, and pensions, where a certain amountof breakeven money needs to be received by the party assuming theunderwritten or contingent risks.

The method of the process for FIG. 5 can also be used to obtain a price,after adjustment for risk, for contingent payoffs for other physical ornatural variables, such as for weather derivatives, where pre-definedpayoffs are functions of observed and measured events in temperature,wind speed, earthquakes, flooding, and other nominal events, or othercatastrophes.

The Wang Price, after discounting, is a useful data result, because itrepresents the present fair value of an asset or liability. This presentfair value can be compared to the present fair value of other financialinstruments, on an even playing field, so that risk managementprofessionals can identify, monitor, acquire, and dispose of assets andliabilities according to portfolio risks and returns.

Example 6 Finding the Fair Value for an Underwritten Underlying RiskVehicle Whose Prospective Future Values can be Either Negative orPositive

For the sixth example of this method, refer to Table 6, which provides aseries of outcomes in 1 year for a special kind of underlying riskvehicle. A risk vehicle can have a wide range of prospective futurecashflow outcomes, including possible negative values, or possiblepositive values for the same future point in time. A blended riskvehicle providing outcomes both of an asset and of a liability, presentsa special difficulty to pricing methods of the prior art.

The risk vehicle in question is a medical insurance policy, for sale onan insurance exchange, with ten known prospective future cashflowoutcomes for the underwritten policy, reflecting various possibilitiesof earned premiums minus incurred losses. Because the underlying is atraded risk vehicle of policy outcomes, the method of the inventionagain turns to FIG. 3, to find the fair value for the collection ofpolicy outcomes at some point in the future. Under the process of thisinvention, for FIG. 3, this pricing is obtained by generating a usefuldata result for the prospective outcomes of the underlying policy,called the Wang Price, as shown in Table 6.

Table 6 provides the series of prospective outcomes in 1 year for amedical insurance policy, in Column 6A. Is the risk vehicle an asset ora liability? One skilled in the art determines that the risk vehicle ofpolicy outcomes represents an asset, because the prospective futurecashflow values, negative and positive, as shown in Column 6A, afterbeing multiplied by their respective probability weights in Column 6B,and summed, into a mean of expected returns, is a positive number. Thispositive number is $6.50. The risk vehicle is thus an asset, and thelambda value calibrating its fair value, will be positive. One skilledin the art understands, however, that if the mean of expected returnswas negative, the risk vehicle would be a liability, and the lambdavalue calibrating any fair value, would be negative.

In FIG. 3, the method starts by determining the objective of theprocess. The objective of the process is to find the Wang Price for thefuture value of the traded risk vehicle, at 301. This satisfies steps301, where the particular future date has been described, by a horizondate of 1 year, and 302, where the selection of a traded underlyinginstrument has been described, by the traded medical insurance policyrisk vehicle.

The medical insurance policy risk vehicle, as underwritten, has adiscounted mean of expected returns of $6.05, because the mean ofexpected returns of $6.50 in 1 year must be reduced by the risk-freeinterest rate of 0.07. But on the insurance exchange, the medicalinsurance policy risk vehicle is being traded at $1.96. This difference,between the mean of future expectations, and the fair value, afteradjustment for risk, is substantial. It reflects the fact that if themedical insurance policy risk vehicle experiences one of the negativelyvalued outcomes, the holder of that risk vehicle must pay out thatoutcome in full to the policyholders.

The method then moves to select a table of prospective future cashflowoutcomes, and their respective probabilities, step 303. The method thenmoves to sort the table of prospective future cashflow outcomes, andtheir respective probabilities, step 304. In Table 6, these steps havealready been taken, in Columns 6A and 6B. When pricing an asset, thelowest cashflows are the largest negative values, and the highestcashflows are the largest positive values, sorted in ascending order.This satisfies steps 303 and 304.

The method then cumulates the probabilities, as shown in Column 6C,satisfying step 305.

The method now selects a lambda value, as the “market price of risk” forthe risk vehicle. The provisional “market price of risk” is calculatedhere by assuming a normal distribution of underwritten income, and usingthe Sharpe Ratio. The expected income for the risk vehicle in 1 year is$6.50, but the market price for this risk vehicle is $1.96. Thus theexpected return for the risk vehicle over 1 year is 2.316, or 231.6% andthe standard deviation of return is 26.295, or 2629.5%. With therisk-free interest rate at 0.07, or 7%, the Sharpe Ratio is 2.316-0.07all divided by 26.295, or 0.0854. This lambda value of 0.0854 satisfiesstep 306.

One skilled in the art follows the steps 307-312, as shown in previousexample, and whose key results are shown in Table 6, as a series oftransformed cumulated values, in Column 6D, a series of decumulatedprobability weights, in Column 6E, and a series of weighted values, inColumn 6F. These weighted values are summed to produce a Wang Price of$2.76, and a weighted Wang Price of $2.58. This satisfies steps 307-312.

At step 313, however, the method requires a decision. Is the Wang Pricefor the risk vehicle close enough to the last quoted market price? Thelast quoted market price for the risk vehicle is $1.96, which is $0.62less than the discounted Wang Price of $2.58. Depending on the toleranceof one skilled in the art for relative lack of precision, the methodallows for application of a payoff function to each variable outcome ofthe underlying, in step 315, or requires further iteration of the lambdavalue, step 314.

For the purposes of this example, the method requires further iterationof the lambda value. The provisional lambda value was 0.0854, based onthe Sharpe Ratio. The method tweaks this provisional lambda value, atstep 314, until the generated Wang Price, from steps 306-312, convergesto equal the last market price of $1.96, step 313. After trial anderror, by one skilled in the art, the calibrated lambda value of 0.10produces a Wang Price of $1.96, as shown in Column 7C of Table 7. Thissatisfies step 313, and completes the first part of the process for FIG.3, at step 312. This completes Example 6.

One skilled in the art understands that the lambda value wasdeliberately iterated so that the Wang Price converged to equal theprevailing market price for the risk vehicle, $1.96. Without aprevailing market price for the risk vehicle, however, the lambda valuemust be inferred or implied from similar risk vehicles, in order to finda fair value.

In accounting, finance, underwriting, and trading, the prior art isdeficient in having a method for providing a fair value, with adjustmentfor risk, for risk vehicles whose overall prospective future cashflowoutcomes were mixes of negative, positive, zero, or infinitesimalnumbers. Any fair value for these mixes is rendered not meaningful. Theinvention, however, calibrates an accurate price for risk vehiclesexperiencing these mixes of negative, positive, zero, or infinitesimaloutcomes. This calibration is a useful data result of the invention.

In accounting, finance, underwriting, and trading, the prior art isdeficient in having a method for providing a fair value, with adjustmentfor risk, for risk vehicles whose history of cashflow outcomes driftedbetween negative, positive, zero, or infinitesimal numbers. Any fairvalue for these drifts is rendered not meaningful. The invention,however, calibrates an accurate price for risk vehicles experiencingthese drifts of negative, positive, zero, or infinitesimal outcomes.This calibration is a useful data result of the invention.

In accounting, finance, underwriting, and trading, the prior art isdeficient in method for providing a fair value, with adjustment forrisk, for risk vehicles that were underwritten, with prospective futurecashflow outcomes based on the future experience of assumed obligations,and that were simultaneously traded, with forward prices exchanged forfuture prospective values. The invention, however, calibrates anaccurate price for risk vehicles that are both underwritten and tradedat the same time.

The Wang Price, after discounting, is a useful data result, because itrepresents the present fair value of an asset or liability. This presentfair value can be compared to the present fair value of other financialinstruments, on an even playing field, so that risk managementprofessionals can identify, monitor, acquire, and dispose of underlyingrisk vehicles, each of which is a group consisting of one or more assetsand liabilities, according to expected portfolio risks and returns.

Example 7 Finding the Fair Value for a Contingent Payoff for anUnderlying Risk Vehicle Whose Prospective Future Values can be Negativeor Positive

For the seventh example of this method, refer to Table 7, which providesa series of outcomes in 1 year for a special kind of underlying riskvehicle, already explored in Example 6. Under the process of thisinvention, by following the steps in FIG. 3, the fair value of the riskvehicle was found to be $1.96, equal to the last market price for therisk vehicle. This fair value was discovered by applying a lambda valuefor the distribution of the underlying risk vehicle, of 0.10, which wassubstantially different than that of the lambda value that would havebeen derived from a Sharpe Ratio.

Table 7 provides the series of prospective outcomes in 1 year for themedical insurance policy, in Column 6A. The risk vehicle has prospectivefuture cashflow outcomes that are both negative and positive. For thisexample, the method calculates the fair value of a put option with astrike price of $0.00. The payoff function for this put option is:

MAX(−Column_(—)6A,0), that is, the maximum of either the negative of aprospective future cashflow value, which itself would be negative, orzero.

In FIG. 3, the method starts by determining the objective of theprocess, step 301. The objective of the process is to find the WangPrice for the value of a put option for the above risk vehicle, with astrike price of 0, in 1 year. This satisfies steps 301, where theparticular future date has been described, by a horizon date of 1 year,and 302, where the selection of a traded underlying instrument has beendescribed, by the traded medical insurance policy risk vehicle. Thisrisk vehicle has already been priced, by the invention, at a fair valueof $1.96.

The method then moves to apply a payoff function to the prospectivefuture cashflow outcomes of the underlying, as found in Column 7C, step309. The results of this payoff function are shown in Column 6A. Forexample, the prospective future cashflow outcome at the top of Column6A, $−123, is taken by the payoff function MAX(−Column_(—)6A,0), toproduce a positive payoff function of $123. This is because the strikeprice of $0, minus the risk vehicle outcome of $−123, is worth apositive $123. With the payoff function applied to all of theprospective future cashflow outcomes, as found in Column 7C, thissatisfies step 315.

The method now multiplies the payoff values to their new probabilityweights, step 316. In Table 7, this means that the distortedprobabilities of Column 7B, are multiplied to the payoff values ofColumn 7D. For example, the distorted probability at the top of Column7B is 0.1187. The payoff value for that probability at the top of Column7D is $123. These are multiplied to produce a weighted payoff at the topof Column 7E, which is $14.60. With all of the weighted payoffs fillingColumn 7E, this satisfies step 316. The method next sums all of theweighted payoffs filling Column 7E, to get an undiscounted Wang Price,step 317. The sum of these weighted payoffs is $17.88, as shown belowthe bottom of Column 7E. This is the undiscounted Wang Price. Thissatisfies step 317.

The method then discounts this Wang Price by the risk-free interestrate, step 318. The risk-free interest rate is 0.07, or 7.00%, annually.The Wang Price of $17.88 is reduced by 7.00% to $16.71. This satisfiesstep 318, and completes the second part of the process for FIG. 3, atstep 314. This completes Example 7.

One skilled in the art would notice that fair value for the put option,at $16.71, is greater than the fair value of the underlying riskvehicle, at $1.46. This reflects the fact that the underlying riskvehicle has a significant degree of prospective negative value, in theform of only-negative outcomes, embedded within the overall slightpositive value, reflecting all of the negative and positive outcomes,averaged together.

In accounting, finance, underwriting, and trading, the prior art isdeficient in having a method for providing a fair value, with adjustmentfor risk, for underlying risk vehicles whose options, contingentpayoffs, or payoff functions, are worth more than the underlying riskvehicles. Any comparative fair value between such an underlying and itsderivative was rendered not meaningful. The invention, however,calibrates an accurate price for underlying risk vehicles whose options,contingent payoffs, or payoff functions, are worth more than theunderlying risk vehicles. This calibration is a useful data result ofthe invention.

Example 8 Finding the Fair Value for an Underlying Risk Vehicle with aRare But Extreme Outcome

For the eighth example of this method, refer to Table 15, which providesa series of two outcomes, success and failure, for a launch of a $200million commercial satellite. The corporate owner of the satellite wantsto purchase an insurance policy to pay $200 million for the destroyedsatellite if the launch fails. To find the fair value of the insurancepolicy, the method follows the steps in FIG. 2, and, for the kernel, orcore process of the Wang Transform, follows the steps in FIG. 6, toemploy a two-factor pricing model.

The first factor in a two-factor pricing model is the lambda value forthe “market price of risk.” The second factor is the “k” number of thedegrees of freedom in a Student-t distribution, for parameteruncertainty in a small sample size.

Table 15 provides the series of prospective future cashflow outcomes forthe satellite launch in Column 15A. The loss amount is $0 for asuccessful launch, and $200 million for an unsuccessful launch. Therespective probabilities for these two prospective outcomes is 0.96 and0.04, as shown in Column 15B. Statistically, the observed populationsize of satellite launches is not very large. Out of 50 launches, only 2have failed.

One skilled in the art can follow the steps 201-205 in FIG. 2, as shownin Column 15C. The method selects a lambda value of −0.2 as the “marketprice of risk” for insuring large equipment losses, like those ofsatellites, based on prevailing market prices for such policies,satisfying step 206. But the rareness of the event, combined with theseverity of the loss, requires a second model factor.

The experience data for satellites is 10,000,000 times more sparse thanthat for auto insurance, and the uncertainty attached to the experiencerequires further compensation. A distribution using “k” degrees offreedom, along with lambda, compensates for this added kind ofuncertainty.

The method then moves to FIG. 6, where the inversion of a standardnormal cumulative distribution is applied to the original probabilityweights, and lambda is added, before a Student-t distribution, with acalibrated number of degrees of freedom, is applied. One skilled in theart can devise many programming approaches to this, but in Excel thefollowing expression may be used, for the entire kernel, or coreprocess, of the two-factor Wang Transform:1−TDIST(NORMINV(COLUMN_(—)15C,0,1)+LAMBDA,K,1).

The TDIST function in Excel returns the Student-t distribution, where Xis the numeric value at which to evaluate the distribution, a range of 1to any arbitrarily large integer, for the number of degrees of freedomto evaluate the distribution, and 1 or 2, for the number of distributiontails to return.

In this expression, the inversion of standard normal cumulativedistribution of the original probability weights, are first shifted bythe lambda value, and then, the Student-t cumulative distribution, asreflected in 1-TDIST, is applied to this expression of shifted weights,with “k” degrees of freedom specified as an integer. The “k” degrees offreedom for this example is the integer 11. This satisfies steps 601-606in FIG. 6. The method now moves back to FIG. 2.

The transformed probabilities are shown in Column 15D. For example, thetransformed probability for no loss is 0.9254. The transformedprobability for a $200 million loss is still 1.000. This satisfies step207.

The method now moves to decumulate these probabilities, step 208. Thedecumulated probability for no loss is 0.9254, as shown at the top ofColumn 15E. The decumulated probability for a $200 million loss is0.0746, as shown at the bottom of Column 15E. This satisfies step 208.

The method next multiplies each prospective future cashflow outcome, asfound in Column 15A, with their respective distorted probabilityweights, as found in Column 15E. The products of these multiplicationsare found in Column 15F. For example, the top loss amount in Column 15Ais $0. The top distorted probability of this loss amount in Column 15Eis 0.9254. The weighted value from multiplying these values is the topweighted value in Column 15F, which is $0.

The bottom loss amount in Column 15A is $200 million. The bottomdistorted probability of this loss amount in Column 15E is 0.0746. Theweighted value from multiplying these values is the bottom weightedvalue in Column 15F, which is $14.93 million. This satisfies step 209.

The method then moves to sum all of the weighted values to get anundiscounted Wang Price, which is $14.93 million, satisfying step 210.This undiscounted price is the same as the discounted price, because thepolicy was obtained the day before the satellite launch, and paid theday after the satellite launch. This satisfies step 211, and closes theprocess in FIG. 2, step 212. This completes Example 8.

The Wang Price, after calibration of lambda, and calibration of the “k”degrees of freedom, and after any needed discounting, is a useful dataresult, because it represents the present fair value of a liability.This present fair value can be compared to the present fair value ofother financial instruments, on an even playing field, so that riskmanagement professionals can identify, monitor, acquire, and dispose ofunderlying risk vehicles comprised of a group of one or more assets andliabilities according to expected portfolio risks and returns.

1. A computer-implemented method for computing and outputting an indicated price, with adjustment for risk, of anticipated contract obligations comprising the steps of: a) identifying an underlying risk vehicle, comprised of a group of one or more assets and liabilities, b) assembling a series of potential future cashflow outcomes, consisting of cashflow values linked to their respectively paired probabilities, as a future probability distribution for that underlying risk vehicle, c) sorting the series of outcomes by their ascending cashflow values, from the lowest listed as first to the highest listed as last, with those cashflow values still linked to their original respectively paired probabilities, d) cumulating the respectively paired probabilities of the sorted series of outcomes so that the last such cumulated probability still linked to the highest cashflow value equals 1, e) providing individual inversely-mapped results for those probabilities, by applying the inversion of the standard normal distribution to all of the cumulated probabilities, f) selecting a lambda value equal to the market price of risk for the overall future probability distribution of the underlying risk vehicle, g) adding the selected lambda value to obtain a shifted inversely-mapped result, h) creating transformed cumulative probability weights, by applying the standard normal cumulative distribution to each shifted result, i) decumulating the transformed cumulative probability weights of the sorted series of outcomes so that the first decumulated weight equals its own cumulated weight, the second decumulated weight equals the second cumulated weight minus the first cumulated weight, the third decumulated weight equals the third cumulated weight minus the second cumulated weight, and so on, continuing until the last decumulated weight equals the last cumulated weight minus the next-to-last cumulated weight, j) producing a set of weighted values, by multiplying the cashflow values to their respective decumulated probability weights; k) computing and outputting an undiscounted future indicated price for the underlying risk vehicle by adding all the weighted values in the set.
 2. The computer-implemented method of claim 1, further comprising the step of discounting the undiscounted price by the risk-free interest rate.
 3. The computer-implemented method of claim 1, further comprising the step of iterating the lambda value, so that the undiscounted price is discounted by the risk-free interest rate, and converges to equal the last recorded outcome or quoted price, of that underlying risk vehicle.
 4. The computer-implemented method of claim 1, further comprising a process of applying a payoff function to each projected cashflow outcome of an underlying risk vehicle, or liability the process comprising the additional steps of: a) applying the payoff function to each projected cashflow outcome of the underlying risk vehicle, b) multiplying the resulting payoff values by their respective decumulated probability weights to produce a set of weighted values, c) adding these weighted values in the set to find an undiscounted price for the payoff function, d) discounting the undiscounted price by the risk-free interest rate.
 5. The computer-implemented method of claim 1, wherein the underlying risk vehicle, or any of the assets and liabilities comprising the underlying risk vehicle, is traded.
 6. The computer-implemented method of claim 5, wherein the assets and liabilities that comprise the underlying risk vehicle are selected from the group consisting of: a) stocks or other equity securities, b) bills, bonds, notes, or other debt securities, c) currencies of various countries, d) commodities of physical, agricultural, or financial delivery, e) asset-backed or liability-linked securities or contractual obligations, f) weather derivatives and other observable physical phenomena whose outcomes can be linked to financial outcomes.
 7. The computer-implemented method of claim 6 wherein securitizations are backed by the assets and liabilities of the underlying risk vehicle.
 8. The computer-implemented method of claim 6 wherein derivatives, contingent claims, and payoff functions, are based on an underlying financial instrument.
 9. The computer-implemented method of claim 6 wherein the underlying risk vehicle comprises part or the whole of the basis of a published benchmark index, or collective experience.
 10. The computer-implemented method of claim 6 wherein the underlying risk vehicle is managed in a portfolio, and in a risk management environment.
 11. The computer-implemented method of claim 6 wherein underlying risk vehicle are managed for the purposes of capital allocation within economic entities.
 12. The computer-implemented method of claim 1, wherein the underlying risk vehicle, or any of the assets and liabilities comprising the underlying risk vehicle, is underwritten.
 13. The computer-implemented method of claim 12 wherein the underwritten underlying risk vehicle, or any of the underwritten assets and liabilities comprising the underlying risk vehicle, are selected from the group consisting of: a) insurance liabilities and reinsurance contracts, b) insurance-linked contracts, and catastrophe bonds, c) credit instruments, including loans, leases, mortgages, and credit cards, d) account payables and receivables.
 14. The computer-implemented method of claim 13 wherein securitizations are backed by the assets and liabilities of the underlying risk vehicle.
 15. The computer-implemented method of claim 13 wherein derivatives, contingent claims, and payoff functions, are based on an underlying risk vehicle.
 16. The computer-implemented method of claim 13 wherein the underwritten risk vehicle comprises part or the whole of the basis of a published benchmark, index, or collective experience.
 17. The computer-implemented method of claim 13 wherein the underlying risk vehicle is managed in a portfolio, and in a risk management environment.
 18. The computer-implemented method of claim 13 wherein underlying risk vehicles are measured for the purposes of capital allocation within economic entities.
 19. The computer-implemented method of claim 13 wherein underlying risk vehicles are measured for cost of capital within economic entities.
 20. The computer-implemented method of claim 1, wherein the (h) step specifying applying a standard normal distribution, is substituted, by applying a distribution selected from the group consisting of: a) a Student-t cumulative distribution, b) the cumulative distribution of a re-scaled Student-t distribution, c) a mixed-normal cumulative distribution, d) a lognormal cumulative distribution, and e) an empirically-constructed cumulative distribution.
 21. A computer-readable medium for use with a computer means for computing and outputting an indicated price, with adjustment for risk, of anticipated contract obligations comprising the steps of: a) identifying an underlying risk vehicle, comprised of a group of one or more assets and liabilities, b) assembling a series of potential future cashflow outcomes, consisting of cashflow values linked to their respectively paired probabilities, as a future probability distribution for that underlying risk vehicle, c) sorting the series of outcomes by their ascending cashflow values, from the lowest listed as first to the highest listed as last, with those cashflow values still linked to their original respectively paired probabilities, d) cumulating the respectively paired probabilities of the sorted series of outcomes so that the last such cumulated probability still linked to the highest cashflow value equals 1, e) providing individual inversely-mapped results for those probabilities, by applying the inversion of the standard normal distribution to all of the cumulated probabilities, f) selecting a lambda value equal to the market price of risk for the overall future probability distribution of the underlying risk vehicle, g) adding the selected lambda value to obtain a shifted inversely-mapped result, h) creating transformed cumulative probability weights, by applying the standard normal cumulative distribution to each shifted result, i) decumulating the transformed cumulative probability weights of the sorted series of outcomes so that the first decumulated weight equals its own cumulated weight, the second decumulated weight equals the second cumulated weight minus the first cumulated weight, the third decumulated weight equals the third cumulated weight minus the second cumulated weight, and so on, continuing until the last decumulated weight equals the last cumulated weight minus the next-to-last cumulated weight, j) producing a set of weighted values, by multiplying the cashflow values to their respective decumulated probability weights; k) computing and outputting an undiscounted future indicated price for the underlying risk vehicle by adding all the weighted values in the set.
 22. The computer-readable medium for use with a computer means of claim 21, further comprising the step of discounting the undiscounted price by the risk-free interest rate.
 23. The computer-readable medium for use with a computer means of claim 21, further comprising the step of iterating the lambda value, so that the undiscounted price is discounted by the risk-free interest rate, and converges to equal the last recorded outcome or quoted price, of that same underlying risk vehicle.
 24. The computer-readable medium for use with a computer means of claim 21, further comprising a process of applying a payoff function to each projected cashflow outcome of an underlying risk vehicle, the process comprising the additional steps of: a) applying the payoff function to each projected cashflow outcome of the underlying asset or liability, b) multiplying the resulting payoff values by their respective decumulated probability weights to produce a set of weighted values, c) adding these weighted values in the set to find an undiscounted price for the payoff function, d) discounting the undiscounted price by the risk-free interest rate.
 25. The computer-readable medium for use with a computer means of claim 21, wherein the underlying risk vehicle is traded.
 26. The computer-readable medium for use with a computer means of claim 25, wherein the assets and liabilities comprising the underlying risk vehicle are selected from the group consisting of: a) stocks or other equity securities, b) bills, bonds, notes, or other debt securities, c) currencies of various countries, d) commodities of physical, agricultural, or financial delivery, e) asset-backed or liability-linked securities or contractual obligations, f) weather derivatives and other observable physical phenomena whose outcomes can be linked to financial outcomes.
 27. The computer-readable medium for use with a computer means of claim 26 wherein securitizations are backed by the assets and liabilities.
 28. The computer-readable medium for use with a computer means of claim 26 wherein derivatives, contingent claims, and payoff functions, are based on the traded assets and liabilities, as underlying financial instruments.
 29. The computer-readable medium for use with a computer means of claim 26 wherein the underlying risk vehicle comprises part or the whole of the basis of a published benchmark, index, or collective.
 30. The computer-readable medium for use with a computer means of claim 26 wherein the underlying risk vehicle is managed in a portfolio, and in a risk management environment.
 31. The computer-readable medium for use with a computer means of claim 26 wherein the underlying risk vehicles are managed for the purposes of capital allocation within economic entities.
 32. The computer-readable medium for use with a computer means of claim 21, wherein the underlying risk vehicle, or any of the assets and liabilities comprising the underlying risk vehicle, is underwritten.
 33. The computer-readable medium for use with a computer means of claim 32 wherein the underwritten underlying risk vehicle, or any of the underwritten assets and liabilities comprising the underlying risk vehicle, are selected from the group consisting of: a) insurance liabilities and reinsurance contracts, b) insurance-linked contracts, and catastrophe bonds, c) credit instruments, including loans, leases, mortgages, and credit cards, d) account payables and receivables.
 34. The computer-readable medium for use with a computer means of claim 33 wherein securitizations are backed by the assets and liabilities of the underlying risk vehicle.
 35. The computer-readable medium for use with a computer means of claim 33 wherein derivatives, contingent claims, and payoff functions, are based on the underlying risk vehicle.
 36. The computer-readable medium for use with a computer means of claim 33 wherein the underwritten risk vehicle comprises part or the whole of the basis of a published benchmark, index, or collective experience.
 37. The computer-readable medium for use with a computer means of claim 33 wherein the underlying risk vehicle is managed in a portfolio, within a risk management environment.
 38. The computer-readable medium for use with a computer means of claim 33 wherein the underlying risk vehicles are measured for the purposes of capital allocation within economic entities.
 39. The computer-readable medium for use with a computer means of claim 33 wherein the underlying risk vehicles are measured for cost of capital within economic entities.
 40. The computer-readable medium for use with a computer means of claim 21, wherein the (h) step specifying applying a standard normal distribution, is substituted, by applying a distribution selected from the group consisting of: a) a Student-t cumulative distribution, b) the cumulative distribution of a re-scaled Student-t distribution, c) a mixed-normal cumulative distribution, d) a lognormal cumulative distribution, and e) an empirically-constructed cumulative distribution. 